Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes
CClassification of the Real Roots of theQuartic Equation and their Pythagorean
Tunes
Emil M. Prodanov
School of Mathematical Sciences, Technological University Dublin,City Campus, Kevin Street, Dublin, D08 NF82, Ireland,E-Mail: [email protected]
Abstract
Presented is a two-tier analysis of the location of the real roots of the general quarticequation x + ax + bx + cx + d = 0 with real coefficients and the classificationof the roots in terms of a , b , c , and d , without using any numerical approximations.Associated with the general quartic, there is a number of subsidiary quadratic equa-tions ( resolvent quadratic equations ) whose roots allow this systematization as wellas the determination of the bounds of the individual roots of the quartic. In manycases the root isolation intervals are found. The second tier of the analysis uses twosubsidiary cubic equations ( auxiliary cubic equations ) and solving these, togetherwith some of the resolvent quadratic equations , allows the full classification of theroots of the general quartic and also the determination of the isolation interval ofeach root. These isolation intervals involve the stationary points of the quartic(among others) and, by solving some of the resolvent quadratic equations , the iso-lation intervals of the stationary points of the quartic are also determined. Thepresented classification of the roots of the quartic equation is particularly useful insituations in which the equation stems from a model the coefficients of which are(functions of) the model parameters and solving cubic equations, let alone using theexplicit quartic formulas, is a daunting task. The only benefit in such cases would beto gain insight into the location of the roots and the proposed method provides this.Each possible case has been carefully studied and illustrated with a detailed figurecontaining a description of its specific characteristics, analysis based on solving cubicequations and analysis based on solving quadratic equations only. As the analysisof the roots of the quartic equation is done by studying the intersection points ofthe “sub-quartic” x + ax + bx with a set of suitable parallel lines, a beautifulPythagorean analogy can be found between these intersection points and the set ofparallel lines on one hand and the musical notes and the staves representing differentmusical pitches on the other: each particular case of the quartic equation has itsown short tune. Mathematics Subject Classification Codes (2020) : 12D10, 26C10.
Keywords : Quartic equation, Cubic equation, Quadratic equation, Roots, Isolationintervals, Pythagorean music. 1 a r X i v : . [ m a t h . G M ] A ug Introduction
The process of solving the general quartic equation x + ax + bx + cx + d = 0 (1)involves the removal of the cubic term by using the substitution x = y − a/
4. Thisresults in the depressed quartic y + py + qy + r = 0, where p = b − (3 / a / , q = c − ( a/ b − a / , and r = d − (3 / a + (1 / a b − (1 / ac . There are severalalgorithms for solving the depressed quartic equation and each of these involves solving acubic equation called resolvent . The resolvents are different for the different algorithms.Finding the roots of the original quartic equation, once the roots of the resolvent cubicare known, is a straightforward procedure in each algorithm.Using the methods presented in [1], this paper presents a two-tier analysis of the loca-tion of the real roots of the general quartic equation (1) with real coefficients and theirclassification.Associated with each quartic, there is a number of subsidiary quadratic equations, re-ferred to in this text as resolvent quadratic equations , with the help of which the roots ofthe general quartic are systematized in terms of its coefficients a, b, c, and d . Addition-ally, the bounds on individual roots are determined. In many cases the root isolationintervals are found, but there is some residual ambiguity as some intervals may containeither two roots of the quartic or no roots at all. The individual root bounds themselvesare associated with the roots of x + ax + b = 0 and cx + d = 0, or with the non-zerostationary points of x ( x + ax + b ), if real roots are absent, or with the points of curva-ture change of x ( x + ax + b ), if real roots and non-zero stationary points are absent, orwith the point of vanishing third derivative of x ( x + ax + b ), if real roots and non-zerostationary points, and curvature change points are absent.Using the proposed classification of the roots of the quartic in terms of the coefficients ofthe quartic and based on solving quadratic equations only would be particularly usefulwhen the quartic results from the study of some model and the coefficients of the quarticare functions of the model parameters. Even the presence of a single parameter in theequation makes the application of the cubic formulas, let alone the quartic formulas,practically impossible and the only benefit would be to get insight into the location ofthe roots.The other tier of the presented analysis is the full classification of the roots of the quarticin terms of its coefficients and, also, the determination of the isolation interval of eachroot of the quartic by solving two cubic equations and some of the resolvent quadraticequations. These two cubic equations are subsidiary to each quartic and are referredto in this paper as auxiliary cubic equations — in order to make a distinction from theresolvent cubic equation.It can be argued why it is necessary to address not one, but two cubic equations (the auxiliary cubic equations ) which do not yield the roots of the quartic, but only revealtheir isolation intervals, and what can be gained by doing so. Indeed, it suffices to solve asingle cubic equation (the resolvent) in order to find the actual roots of any quartic. The2nswer to this is in the search for rules and patterns through abstraction to gain insighton how different coefficient vectors affect the roots. Systematization and having predic-tive powers are always a bonus and merit investigation. If analysis of the roots of thequartic could be done with equations of degree two, one would expect that using equa-tions of degree three would be more informative, despite getting into the realm wherethe procedure of finding the roots through the explicit formulas is applied. The analysisbased on solving cubic equations complements the picture revealed by analysis based onquadratic equations. The resulting systematization and classification are not possibleif one addresses the explicit formulas for the roots of the quartic — these are ratherunwieldy and one cannot trace how the variation of a particular coefficient of the givenquartic affects the roots, i.e. the coefficients of the quartic enter the root formulas in anintricate combination involving the root(s) of the resolvent cubic (which, in turn, dependon the coefficients of the quartic) and one cannot discern the individual contribution ofeach coefficient of the quartic to the location of its roots. The proposed analysis basedon cubic equations also has heuristic potential: just by observation of the coefficientsand whether they fall into specific ranges, one can predict the number of real roots andalso find their isolation intervals. For example, for any a, c, and d , when b > (3 / a , thequartic cannot have four real roots. If, additionally, the free term d is negative and c isalso negative, then the quartic has one negative root smaller than − d/c and one positiveroot greater than the only real non-zero root λ > d . If µ denotes the only stationary point of quartic( µ > d is positive and greater than µ + aµ + bµ + cµ (which itselfis positive), then the quartic has no real roots. If 0 < d < µ + aµ + bµ + cµ , then thequartic has two positive roots — one bigger than − d/c and smaller than µ , the other –bigger than µ and smaller than λ .In the presented analysis, the isolation intervals of the stationary points of the generalquartic are also determined — with the help of the resolvent quadratic equations .The analysis of the roots of the quartic equation is done after studying the intersectionpoints of the “sub-quartic” x + ax + bx with a set of suitable parallel lines. A beautifulPythagorean analogy exists between these intersection points and the set of parallel lineson one hand and the musical notes and the staves representing different musical pitcheson the other. Even more, each case of the quartic equation has its own tune.Every possible situation has been individually studied and illustrated with a detailedfigure containing a description of its specific characteristics, analysis based on solvingcubic equations and analysis based on solving quadratic equations only. Each quartic is associated with a number of subsidiary cubic, quadratic, and linearequations whose roots can be used for the classification of the roots of the quartic.In parallel with the presentation of the analysis, Figures 1 to 5 illustrate, through aparticular example with the quartic x + x − x − x + 1, the full procedure of findingthe isolation interval of each root of the quartic, based on solving cubic equations, and3he localization of the roots by determination of the individual root bounds, based onsolving quadratic equations only. The isolation intervals of the stationary points of thisquartic are also found.Figures 6 to 12 illustrate some patterns associated with the general quartic and theseare used for the classification of the roots of the quartic.Taking the free term d of the quartic and varying it, yields a one-parameter congruenceof quartics, all having the same set of stationary points (which are either three — twolocal minima and a local maximum or a saddle point and a local minimum, or justone — a minimum). Let µ i denote the stationary point(s). For each µ i , there is a“special” quartic within this congruence: x + ax + bx + cx + δ i — the one whose graphis tangent to the abscissa at that particular stationary point. The derivative of the“special” quartic is also zero at this stationary point, namely, for x + ax + bx + cx + δ i the stationary point µ i is also a double root [or a triple root, if the original quartichas a saddle at µ i , or a quadruple root − a/ x − a/ = x + ax + (3 / a x + (1 / a x + (1 / a , which coincides with its only “special”quartic]. Setting the derivative of the quartic equal to zero yields the set of its stationarypoints and the resulting equation,4 x + 3 ax + 2 bx + c = 0 , (2)is referred to in this text as first auxiliary cubic equation .Substituting each real root µ i of this equation into the corresponding “special” quarticequation x + ax + bx + cx + δ i = 0, immediately gives: δ i = − µ i − aµ i − bµ i − cµ i . (3)Thus the “special” quartics are given by x − µ i + a ( x − µ i ) + b ( x − µ i ) + c ( x − µ i ).The discriminant of the first auxiliary cubic equation is ∆ = − c − a ( a / − b ) c + 128 b [(9 / a / − b ]. It can be viewed as a quadratic in c , treated as unknown,with a and b treated as parameters. The first resolvent quadratic equation is obtainedby setting ∆ = 0: c + a (cid:32) a − b (cid:33) c − b (cid:32) a − b (cid:33) = 0 . (4)The roots of this equation, c , ( a, b ) = c ± √ (cid:115)(cid:18) a − b (cid:19) , with c ( a, b ) = 12 a (cid:32) b − a (cid:33) , (5)play a very important role in the analysis. For any given quartic, one has to see firstwhether the coefficient c of the linear term falls between the roots c , ( a, b ) or outsidethem. If c ( a, b ) ≤ c ≤ c ( a, b ), then the discriminant ∆ is positive and the quartichas three stationary points: µ , , . Otherwise it has just one: µ . In the first case, the4uartic can have either 0, or 2, or 4 real roots; in the second case it can have either 0 or2 real roots. It is immediately obvious that, for any a, c, and d , when b > (3 / a / ∆ of the first auxiliary cubic equation (2) is equal to zero, thatis, when c = c , ( a, b ), the original quartic with c replaced by c , has a saddle pointat η , and a local minimum at θ , . The corresponding “special” quartic at η , is x + ax + bx + c , x + d , , where d , = − η , − aη , − bη , − c , η , , and for the“special” quartic, η , is a triple root. The points η , and θ , can be easily found asthe “special” quartic x + ax + bx + c , x + d , , its first derivative, and its secondderivative are all zero at η , . In other words, one has to start with solving the secondresolvent quadratic equation , 6 x + 3 ax + b = 0 , (6)the roots of which are η , = − a ± √ (cid:115) a − b, (7)then write down the vanishing first derivative of the (“special”) quartic as 4 x + 3 ax +2 bx + c , = 4( x − η , ) ( x − θ , ), and then compare the coefficients of the quadraticterms. This will give θ , = − (3 / a − η , and hence: θ , = − a ± √ (cid:115) a − b. (8)One could observe that − a/ x + ax + (3 / a x +(1 / a x + (1 / a = 0.With the help of η , and θ , , the isolation intervals of the stationary points µ i of thegeneral quartic can be easily found (see the example on Figure 6). In the regime ofincreasing c and starting with c < c , there is only one stationary point (local minimum)at (cid:98) µ > θ . When c = c , the quartic has a saddle point at η and a local minimum at θ . As soon as c gets bigger than c , the saddle point η bifurcates into two stationarypoints µ and µ on either side of η : a local maximum at µ such that η < µ < η and a local minimum at µ such that θ < µ < η . The local minimum (cid:101) µ remains as µ and is such that η < µ < θ . With the further increase of c , the local maximum at µ and the right local minimum (at µ ) get closer to each other and coalesce at η when c = c . The left local minimum is then at θ . This corresponds to a saddle point η anda local minimum at θ for the quartic with c = c . When c becomes bigger than c , thequartic will have only one stationary point — the local left local minimum µ remainsas (cid:101) µ < θ .To summarize, the isolation intervals of the stationary points of the general quartic areas follows (dropping the tilde and the hat): (i) If c < c , the quartic has a single local minimum µ > θ .5 ii) If c = c , the quartic has a saddle point at η and a local minimum at θ . (iii) If c < c < c , the quartic has a local minimum at µ where θ < µ < η , alocal maximum at µ where η < µ < η , and local minimum at µ where η < µ < θ . (iv) If c = c , the quartic has a saddle point at η and a local minimum at θ . (v) If c > c , the quartic has a single local minimum µ < θ .For the analysis further, one also needs to determine the other two roots ξ ( i )1 , of the“special” quartics x + ax + bx + cx + δ i (recall that µ i is at least a double root forthem). One has: x + ax + bx + cx + δ i = ( x − µ i ) ( x − ξ ( i )1 )( x − ξ ( i )2 ) = 0 . (9)Vi`ete formulas give: ξ ( i )1 + ξ ( i )2 = − a − µ i and 2 µ i [ ξ ( i )1 + ξ ( i )2 ] + µ i + ξ ( i )1 ξ ( i )2 = b . Fromthese one finds that ξ ( i )1 ξ ( i )2 = b + 3 µ i + 2 aµ i . If x (cid:54) = µ i then (9) reduces to x + ( a + 2 µ i ) x + b + 3 µ i + 2 aµ i = 0 . (10)This is the third resolvent quadratic equation . The roots of this equation are: ξ ( i )1 , = − a − µ i ± (cid:113) a − aµ i − µ i − b. (11)Note that the roots of the third resolvent quadratic equation (10) depend on the roots µ i of the first auxiliary cubic equation (2). That is, to find the ξ ’s, one needs to findat least one of the stationary points of the quartic. Thus, the third resolvent quadraticequation should be used in the analysis based on solving cubic equations.Separately, for one of the three “special” quartics x + ax + bx + cx + δ i , the roots ξ ( i )1 , of the third resolvent quadratic equation (10) are not real, while for the remainingtwo they are real (unless the original quartic has the same value at its two local minima,in which case two of the “special” quartics x + ax + bx + cx + δ i coincide and so all“special” quartics will have four real roots) — see Figure 2.In the congruence of quartics, there is another significant quartic — the one that passesthrough the origin — i.e. this is a privileged quartic as it is the only one that has zeroas a root. It is obtained from the original quartic by removing the free term d . Theremaining three roots of this privileged quartic are found by solving the second auxiliarycubic equation : x + ax + bx + c = 0 . (12)If the discriminant ∆ = − c + ( − a + 18 ab ) c + a b − b of this equation is negative,there is only one real root: λ . If it is not negative, there are three real roots: λ , , .To determine which of these occurs, set ∆ = 0 to obtain the fourth resolvent quadraticequation : c + 23 a (cid:32) a − b (cid:33) c − b (cid:18) a − b (cid:19) = 0 . (13)6nce again, setting a discriminant of a cubic equal to zero is viewed as a quadratic inthe unknown c with a and b treated as parameters. The roots of this equation, γ , ( a, b ) = 13 a (cid:32) b − a (cid:33) ± √ (cid:115)(cid:18) a − b (cid:19) . (14)also play a very important role in the analysis. If the given c is such that γ ( a, b ) ≤ c ≤ γ ( a, b ), then the quartic x + ax + bx + cx has four real roots: 0 and λ , , (there maybe zeros among the λ ’s). Otherwise, the x + ax + bx + cx has only two real roots:zero and λ (which may also be zero).Following the ideas of [1], the four “degrees of freedom” of the general quartic x + ax + bx + cx + d are split equally between two separate polynomials, x + ax + bx and − cx − d , the difference of which comprises the given quartic and the “interaction”between which gives the roots of the quartic: x ( x + ax + b ) = − cx − d (15)— see Figure 3 (and also Figures 4 and 5) where this is illustrated with an example.It may be tempting to depress the quartic and analyse only its “three-dimensional pro-jection” y + py + qy + r = 0, but by doing so, study of how the coefficients of the originalquartic affect its roots would not be possible as these coefficients would be “dissolved”into p , q , and r .It is quite easy to analyse the two parts of the “split” quartic and, hence, the quarticitself — one of the “components” is a straight line, while the other is a quadratic “indisguise” — in the sense that it is a quartic having zero as a double root and allowinganalysis not more difficult than that of a genuine quadratic (with the possible additionof a pair of stationary points and/or a pair of curvature change points, and the additionof a point where the third derivative vanishes).For any given a and b , i.e. for any “sub-quartic” x ( x + ax + b ), one can find a straightline − c † x − d † such that it will be tangent to x ( x + ax + b ) at two points, say α and β . This means that the obtained in this manner quartic, x + ax + bx + c † x + d † , hastwo double roots: α and β . That is, x + ax + bx + c † x + d † = ( x − α ) ( x − β ) = x − α + β ) x + ( α + β + 4 αβ ) x − αβ ( α + β ) x + α β . Comparing the correspondingcoefficients yields: c † = (1 / a ( b − a / c — see the roots (5)of the first resolvent quadratic equation (4). One also gets d † = (1 / b − a / > c and d will be used instead of c † and d † .The quartic x + ax + bx + c x + d is very important for the classification of the rootsof the general quartic. This is better visualized on the tablecloth of the “split” quartic(15). The determination of whether c is smaller, equal, or greater than c will determinethe ordering of the δ ’s and will also determine the number of intersection points betweenthe “sub-quartic” x ( x + ax + b ) and − cx − d for any value of d . For example, on Figure5, one has 0 < c = − . < c = −
1. Thus, if one studies the intersection points of x ( x + ax + b ) with − cx − d in the regime of increasing ( − d ) starting from −∞ , i.e.“sliding” a straight line with fixed slope ( − c ) upwards, intersections of this straight line7ith x ( x + ax + b ) will occur first in the third quadrant before they occur in the forth.From the roots (5) of the first resolvent quadratic equation (4), it is immediately obviousthat c ( a, b ) ≤ c ( a, b ) ≤ c ( a, b ) for any a and b . The graph of c as a function of a and b is shown on Figure 7. Figures 8 to 11 show that, for any a and b , the following holdsfor the general quartic: c ( a, b ) ≤ γ ( a, b ) ≤ c ( a, b ) ≤ γ ( a, b ) ≤ c ( a, b ) . (16)Depending on a and b , the place of zero in the above chain of inequalities could beanywhere. For the analysis of the general quartic, the very first step is the determinationof these numbers and the following step is to find the place of 0 and the given c in theabove. Figure 12 shows this chain when a and b are both negative, in which caseone has c < γ < < c < γ < c . On Figure 12, the “separator” straight lines − c , x − d , , − γ , x , and − c o x − d are drawn and this clearly demonstrates that(including also the coordinate axes) there are seven ranges in which c may fall. Each ofthese is individually studied. It has its own peculiarities that reflect on the number ofroots and their localization.Next, for the classification of the roots based on solving cubic equations, one needs tofind the place of ( − d ) among the ( − δ )’s and zero — see Figure 4 where this is illustratedwith the quartic equation x + x − x − x + 1 = 0. The role of the second auxiliarycubic equation (12) and its roots λ , , now becomes clear. Keeping a and b fixed [i.e.not changing the “sub-quartic” x ( x + ax + b )], and only varying c would “move” thestationary points µ i along the “sub-quartic” x ( x + ax + b ). For the example with a = 1 and b = − c <
0, one always has − δ > x ( x + ax + b ). On the other hand however, − δ and − δ could beanywhere. Because the second auxiliary cubic equation (12) has three real roots λ , , , − δ and − δ are both negative — these are on the “other side” (opposite side of − δ ) ofthe straight line − cx − x + ax + bx + cx ).If the second auxiliary cubic equation (12) had just one root λ , then − δ and − δ wouldboth be positive. And because c > c , one has − δ < − δ .For the other tier of the analysis — based on solving quadratic equations only — onedoes not have the µ ’s, the δ ’s, and the λ (cid:48) s explicitly. The isolation intervals of the δ ’sand the λ ’s can be found in a manner similar to the one used for the determinationof the isolation intervals of the µ ’s or one can see [1] for the full classification of theroots of the cubic equation and the determination of the isolation intervals of its roots.The “separator” line − cx can still be used without knowing the loci of its point(s) ofintersection with the “sub-quartic” x ( x + ax + b ), but knowing if these are three orjust one. The “separator” lines − cx − δ i can no longer be used for analysis based onsolving quadratic equations only. There is a way however, to find “replacements”.The “sub-quartic” x ( x + ax + b ) has zero as a double root and two more roots whichare the roots of the fifth resolvent quadratic equation x + ax + b = 0 , (17)8he roots of which are ρ , = − a ± (cid:115) a − b. (18)These are real for b ≤ a / c and draws the two parallel lines with equations − c ( x − ρ , ).These straight lines are the sought “replacements” of the “separator” lines − cx − δ i andtheir intersections with the ordinate — the “marker” points cρ , — are the “replace-ments” of the ( − δ )’s. This allows the analysis based on solving quadratic equations onlyto be performed in manner fully analogous to that of the analysis based on solving cubicequations — see Figure 5.All possibilities for this analysis are shown on Figures 1.1 to 1.14 and 2.1 to 2.14.If b > a / ρ , not being real), a different pair of characteristic points of the “sub-quartic” x ( x + ax + b ) should be chosen as “marker” points — the two non-zerocritical points σ , of x ( x + ax + b ). Clearly, recourse to these can be made for a / < b ≤ (9 / a / sixth resolvent quadratic equation x + 3 ax + 2 b = 0 , (19)the roots of which are the non-zero critical points of x ( x + ax + b ) given by σ h,H = − a ± √ (cid:115) a − b. (20)One then calculates the values H and h of x ( x + ax + b ) at σ H and σ h respectively anddraws the parallel lines − c ( x − σ H ) + H and − c ( x − σ h ) + h to serve as “separators”.These intersect the ordinate at the “marker” points cσ H + H and cσ h + h . One has to becareful because, depending on c , one can have zero, cσ H + H , and cσ h + h in any order.All possibilities for this analysis are shown on Figures 3.1 to 3.14.Should b be greater than (9 / a / x ( x + ax + b ) will nothave critical points. One should then use the points of curvature change [non-zero firstderivative, but vanishing second derivative of the “sub-quartic” x ( x + ax + b )]. Theseare real for b ≤ (3 / a / second resolvent quadratic equation and on Figures 4.1 to 4.14 and, also 5.1 to 5.10 (where the relevant analysis is), they aredenoted by τ h,H = − a ± √ (cid:115) a − b. (21)As with the critical points σ , , one then calculates the values H and h of x ( x + ax + b )at τ H and τ h respectively and draws the parallel lines − c ( x − τ H ) + H and − c ( x − τ h ) + h to serve as “separators”. These intersect the ordinate at the “marker” points cτ H + H and cτ h + h . The slope of the straight line joining the two points of curvature changeis equal to ± c . Thus, which of cτ H + H and cτ h + h is bigger depends on whether c is9igger or smaller than c . Again, care should be exercised as zero, cτ H + H and cτ h + h could be in any order — it is the number of real roots of the second auxiliary cubicequation that determines this order.Finally, when b > (3 / a /
4, not one of the resolvent quadratic equations has real roots.One still needs to find identifiable points the “sub-quartic” x ( x + ax + b ) from which“separator” lines can be drawn. There is just one such point — where the third derivativeof x ( x + ax + b ) vanishes. This point is the only root φ = − a/ resolventlinear equation : 4 x + a = 0. One then draws the only available “separator” — the line − c ( x + a/
4) + (1 / a [ b − (3 / a / t = − (1 / ac + (1 / a [ b − (3 / a / a and c are opposite, a second“separator” line, − cx + (1 / a [ b − (3 / a / a and c are with the same sign).Care should be exercised as one could have t < , t = 0, or t > Every possible case for non-zero a , b , and c has been thoroughly analyzed. The cases of c equal to c , , or γ , , or c do not get special attention either — should one or more of a , b , and c be zero or should c be equal to one of the above, the analysis (not presentedhere) follows trivially.The results of the investigation are presented in figures with labels i.j , where i and j are positive integers . The figures can be grouped into a (rather large) table. Forease of reference, an effort has been made to keep the individual figures independentfrom each other and for this purpose, each Figure i.j contains a short description ofthe situation, analysis based on solving cubic equations, and analysis based on solvingquadratic equations only.The index i in Figure i.j runs from 1 to 6 and labels the rows of the table: i = 1 : This is the case of b <
0. The roots c , , γ , , and ρ , of the first, fourth,and fifth resolvent quadratic equation , respectively, are real. The roots ρ , have oppositesigns. When i = 1, the index j runs from 1 to 14 (there are fourteen columns). The firstseven of these correspond to the seven possible ranges for c when a <
0; the remainingseven are the seven possible ranges for c when a > i = 2 : This is the case of 0 < b ≤ a /
4. The roots c , , γ , , and ρ , of the first,fourth, and fifth resolvent quadratic equation , respectively, are again real. This time theroots ρ , have the same sign (opposite to the sign of a ). When i = 2, the index j againruns from 1 to 14 (there are fourteen columns) with the first seven of these correspondingto the seven possible ranges for c when a < c when a > i = 3 : This is the case of a / < b ≤ (9 / a /
4. The roots c , and γ , of the first and fourth resolvent quadratic equation , respectively, are real, but the roots ρ , ofthe fifth resolvent quadratic equation , are not real. The roots σ h,H of the sixth resolvent uadratic equation are real and these are used in the analysis. When i = 3, the index j again runs from 1 to 14 (there are fourteen columns). The first seven of these correspondto the seven possible ranges for c when a <
0; the remaining seven are the seven possibleranges for c when a > i = 4 : This is the case of (9 / a / < b ≤ (4 / a /
4. The roots c , and γ , ofthe first and fourth resolvent quadratic equation , respectively, are real, but the roots ρ , and σ h,H of the fifth and sixth resolvent quadratic equation , respectively, are not real.The roots τ h,H of the second resolvent quadratic equation are real and these are usedin the analysis. When i = 4, the index j again runs from 1 to 14 (there are fourteencolumns). The first seven of these correspond to the seven possible ranges for c when a <
0; the remaining seven are the seven possible ranges for c when a > i = 5 : This is case of (4 / a / < b ≤ (3 / a /
4. The roots c , of the firstresolvent quadratic equation are real, but the roots γ , , ρ , , and σ h,H , of the fourth,fifth, and sixth resolvent quadratic equation , respectively, are not real. The roots τ h,H ofthe second resolvent quadratic equation are real and these are again used in the analysis.When i = 5, the index j runs from 1 to 10 (there are ten columns). The first five ofthese correspond to the five possible ranges for c when a <
0; the remaining five are thefive possible ranges for c when a > γ ’s anymore. i = 6 : This is the case of (3 / a / < b . Not one of the resolvent quadraticequations has real roots. But the graph of the “sub-quartic” x ( ax + bx + c ) still hasa “blemish” that can be used and extract one or two “separator” lines for the analysis.This is the point φ = − a/ x ( ax + bx + c ) is zero. i.e. φ is the root of the resolvent linear equation x + a = 0. When i = 6, the index j runsfrom 1 to 4 (there are only four columns): a < c < , a < c > , a > c <
0, and a > c > b of the quadratic term falls into. The relevant row of the table is then selected by theparticular value of b .Next, if the sign of the coefficient a of the cubic term is negative, one should look atcolumns 1 to 7 for the first four rows of the table, columns 1 to 5 for the fifth row andcolumns 1 and 2 for the sixth row of the table. If a is positive, one should look at theother columns of the relevant row.It is the place of the coefficient c of the linear term within the chain of inequalities (16)that determines which particular column applies, namely, c selects the individual cell inthe table that is relevant.Finally, the value of the coefficient d of the free term determines, within that cell, whichone of the cases labeled by lower case Roman numerals applies (either for the analysisbased on cubic equations or for the analysis based on solving quadratic equations only).All figures are on pages 12–61. 11 igure 1 Figure 2 The quartic polynomial x + x − x − x + 1 has four realroots x i and three stationary points: a local minimum at µ = 1 .
00, a local maximum at µ = − .
16, and anotherlocal minimum at µ = − .
59 — roots of the first auxiliarycubic equation: 4 x + 3 x − x − . (The second auxiliary cubic equation is x + x − x − λ , , — see also Figure 2 andFigure 4.) Shown are the significant quartics of the congruence ob-tained by varying the free term of the original quartic x + x − x − x + 1 (also pictured): the three “special”quartics: x + x − x − x + 2 . , x + x − x − x − . , and x + x − x − x + 3 . , together with the quartic withouta free term, x + x − x − x. Figure 3
The quartic polynomial x + x − x − x + 1 is viewed as thedifference of two polynomials: f ( x ) = x ( x + x −
3) and g ( x ) = x −
1. The intersection points of f ( x ) and g ( x ) are the roots ofthe given quartic equation x + x − x − x + 1 = 0. The quartic f ( x ) has a double root at zero and two more roots: ρ = − . ρ = 1 .
30. The straight line g ( x ) has crosses the abscissa at x = 1.The roots of the quartic equation x + x − x − x + 1 = 0 arestudied in a two-tier analysis — by solving cubic equations (seeFigure 4) and by solving quadratic equations only (see Figure 5).This equation (with a = 1, b = −
3, and c = −
1) belongs to thesub-case shown on Figure 1.11. igure 4 Analysis based on solving cubic equations
The equation x + x − x − x +1 = 0 comes with c = − c = − .
00 and c = 1 .
75 of the first resolventquadratic equation (4). Thus, the first auxiliary cubic equation (2) has three real roots, i.e. the quartic has three stationary points( µ = 1 . , µ = − . , µ = − .
59) and can have 0, 2 or 4 realroots.As c = − γ = − .
42 and γ = 1 . fourth resolvent quadratic equation (13), the second aux-iliary cubic equation (12) has three real roots ( λ = 1 . , λ = − . , λ = − . x − x ( x + x − − δ < − δ <
0. On the other hand, − δ > c (cid:54) = 0.As 0 < c = − . < c (see Figure 5 where the line − c x − d isshown), one has − δ < − δ .The real roots ξ (1 , , of the third resolvent quadratic equation (10)are those corresponding to µ , .The line x − ρ = 1 . d = 1 satisfies − δ = − . < − δ = − . < − d < < − δ = 0 .
08, the number of real roots is exactly 4.Therefore, there is a negative root x between λ and ξ (1)2 , anegative root x between ξ (1)1 and λ , a positive root x smallerthan min { µ , − d/c } , and a positive root x between µ and λ (a sharper bound for x is ρ < x < λ ). See also Figure 1.11. Figure 5
Analysis based on solving quadratic equations only
To perform this analysis, it is necessary first to determine, bysolving the first and third resolvent quadratic equations , all num-bers in the chain of inequalities c ≤ γ ≤ c ≤ γ ≤ c (see alsoFigures 7 to 12) and then find in this chain the places of 0 andthe given c = − a = 1 and b = −
3, one finds: c = − < γ = − . < c = − . < c = − < < γ = 1 . < c = 1 . d = 2 .
64 and, by solving another quadratic equa-tion, α = 1 .
05 and β = − . c being between c and c , the given quartic equationcould have 0, 2, or 4 real roots.Due to c being between γ and γ , there are three intersectionpoints ( λ , , ) of x + ax + bx with − cx .Due to c > c , it is guaranteed that there are two intersectionpoints of x + ax + bx with − cx − d in the third quadrant,provided that − d > cρ .For the given a , b , and c , the analysis shown on Figure 1.11 applieswith (ii) being the relevant sub-case, namely, cρ = − . ≤− d = − <
0. Thus there are four real roots: the two negativeroots x , > ρ = − .
30, the positive root x < − d/c = 1 andthe positive root x > ρ = 1 . igure 6 Isolation Intervals of the Stationary Points of the General Quartic
The example chosen here for illustration is a quartic with coefficients: a = 1, b = − c = −
1, and d = 1.The first auxiliary cubic polynomial 4 x + 3 ax + 2 bx + c is plotted on the left pane with fixed a = 1 and b = − c varyingin different ranges, determined by the roots c = − .
41 and c = 4 .
16 of the first resolvent quadratic equation (4). The quartic x + ax + bx + cx + d is plotted on the right pane, again with fixed a = 1 and b = −
5, also with fixed d = 1, and with c varyingover the same ranges as on the right pane. One can immediately see how the coefficient c in the linear term affects the quartic anddetermine the number and type of stationary points of the quartic.By solving the second resolvent quadratic equation , one immediately finds that η = 0 .
70 and η = − .
20. By using (8), one findsthat θ = − .
14 and θ = 1 . (i) If c < c , the quartic has a single local minimum (cid:98) µ > θ (purple solid curve). (ii) If c = c , the quartic has a saddle point at η and a local minimum at θ (blue dashed curve). (iii) If c < c < c , the quartic has a local minimum at µ with θ < µ < η , a local maximum at µ with η < µ < η , and localminimum at µ with η < µ < θ (black solid curve). (iv) If c = c , the quartic has a saddle point at η and a local minimum at θ (red dash-dotted curve). (v) If c > c , the quartic has a single local minimum (cid:101) µ < θ (green solid curve).Thus − . < µ < − . < µ < . < µ < .
64. The actual values are: µ = − . µ = − . , and µ = 1 . Figure 7 Figure 8 c ( a, b ) = a (cid:16) b − a (cid:17) . γ ( a, b ) = a (cid:16) b − a (cid:17) + √ (cid:113)(cid:0) a − b (cid:1) ≥ c ( a, b ) = a (cid:16) b − a (cid:17) . igure 9 Figure 10 γ ( a, b ) = a (cid:16) b − a (cid:17) + √ (cid:113)(cid:0) a − b (cid:1) ≤ c ( a, b ) = c ( a, b ) + √ (cid:113)(cid:0) a − b (cid:1) . γ ( a, b ) = a (cid:16) b − a (cid:17) − √ (cid:113)(cid:0) a − b (cid:1) ≤ c ( a, b ) = a (cid:16) b − a (cid:17) . Figure 11 Figure 12 c ( a, b ) = c ( a, b ) − √ (cid:113)(cid:0) a − b (cid:1) ≤ γ ( a, b ) = a (cid:16) b − a (cid:17) − √ (cid:113)(cid:0) a − b (cid:1) . Pictured here are the “sub-quartic” x ( x + ax + b ) and the“separator” straight lines − c , x − d , , − γ , x , and − c x − d when a and b are both negative (in which case c is positive).On the diagram, η , are the roots of the second resolventquadratic equation , i.e. the triple roots of the “special” quartics x + ax + bx + c , x + d , .For any a and b , one always has c ≤ γ ≤ c ≤ γ ≤ c .For any particular pair ( a, b ), the place of zero has to be found inthis chain of inequalities and then one has to determine in whichof the seven ranges (determined by the “separator” lines and thecoordinate axes) the coefficient c of the linear term falls. igure 1.1 Figure 1.2 b < , a < , c < , b < , a < , c < ,c < c < γ < < c < γ < c c < c < γ < < c < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ < ρ (pictured) or µ ≥ ρ .Obviously, − δ ≤ cρ < < cρ .The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x ≤ min { µ , − d/c } and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d , then there is a non-positive root x such thatmin { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots (these are greater than ρ if µ > ρ or smallerthan or equal to ρ if µ ≤ ρ ). (ii) If cρ ≤ − d < x < − d/c and another positive root x ≥ ρ . (iii) If 0 ≤ − d < cρ , then there is one negative root x such that ρ < x ≤ − d/c and a positive root x > ρ . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and a positive root x > ρ . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ > − δ < − δ < cρ < < − δ < − δ < cρ .The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x ≤ min { µ , − d/c } and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d < − δ , then there is a positive root x such that λ ≤ x < ξ (3)1 , and a negative root x such that ξ (3)2 < x ≤− d/c . (iv) If − δ ≤ − d < − δ , then there is a positive root x suchthat ξ (3)1 ≤ x < ξ (2)1 , a negative root x such that µ < x ≤ min { ξ (3)2 , − d/c } , another negative root x such that µ ≤ x <µ , and a third negative root x such that µ ≤ x < ξ (2)2 . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots x , < ρ . (ii) If cρ ≤ − d < x < − d/c and another positive root x ≥ ρ . (iii) If 0 ≤ − d < cρ , then there are either three or one negativeroots greater than ρ and smaller than or equal to − d/c and therealso is one positive root greater than ρ . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and a positive root x > ρ . igure 1.3 Figure 1.4 b < , a < , c < , b < , a < , c > ,c < γ < c < < c < γ < c c < γ < < c < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − δ < cρ ≤ − δ < < − δ < cρ or − δ < − δ < cρ < < − δ < cρ (pictured).The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is a positive root x such that ξ (3)2 < x ≤ µ and another positive root x such that µ ≤ x <ξ (3)1 . (iii) If − δ ≤ − d < x such that λ < x ≤ µ , another negative root x such that µ ≤ x < λ , a positive root x < min { ξ (3)2 , − d/c } and anotherpositive root x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , another negative root x such that λ ≤ x < µ ,a third negative root x such that µ < x < − d/c , and a positiveroot x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or there aretwo positive roots x , < ρ , or there are two two positive roots x , < ρ together with two negative roots x , > ρ (the latterappear when cρ ≥ − δ ). (ii) If cρ ≤ − d < x < − d/c and x ≥ ρ , or there are two negative roots x , > ρ together with two positive roots: x < − d/c and x ≥ ρ . (iii) If 0 ≤ − d < cρ , then there are either three negative roots x , , such that ρ < x , , ≤ − d/c together with a positive root x > ρ , or there is one negative root x such that ρ < x ≤− d/c and a positive root x > ρ . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and one positive root x > ρ . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also − δ < − δ > − c x − d with c = (1 / a ( b − a / > d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As 0 < c < c , one has − δ < − δ . Thus − δ < − δ < cρ < < − δ < cρ .The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x suchthat ξ (3)2 < x ≤ µ and another positive root x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d < x such that λ < x ≤ µ , another negative root x ≥ max {− d/c, µ } , a positive root x such that λ < x ≤ ξ (3)2 andanother positive root x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ and three positive roots: x such that − d/c ≤ x < µ , x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or thereare two positive roots x , < ρ , or there are two negative roots x , > ρ together with two positive roots x , < ρ . (ii) If cρ ≤ − d < x ≤ ρ , another negative root x > − d/c , and two positive roots x , < ρ . (iii) If 0 ≤ − d < cρ , then there is one negative root smaller than ρ and either one or three positive roots greater than or equal to − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ and apositive root x such that ρ ≤ x < − d/c . igure 1.5 Figure 1.6 b < , a < , c > , b < , a < , c > ,c < γ < < c < c < γ < c c < γ < < c < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − c x − d with c = (1 / a ( b − a / > d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c > c , one has − δ < − δ . Thus one can have − δ < cρ ≤− δ < < − δ < cρ or − δ < − δ < cρ < < − δ < cρ (pictured).The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (1)2 < x ≤ µ and another negative root x such that µ ≤ x < ξ (1)1 . (iii) If − δ ≤ − d < x such that λ < x ≤ ξ (1)2 , another negative root x ≥ max {− d/c, ξ (1)1 } a positive root x such that λ < x ≤ µ andanother positive root x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x suchthat ξ (2)2 < x ≤ λ and three positive roots x , , such that: − d/c ≤ x < µ , µ < x ≤ λ , and λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or there aretwo negative roots x , > ρ , or there are two negative roots x , > ρ and two positive roots x , < ρ (the latter appearwhen cρ ≥ − δ ). (ii) If cρ ≤ − d < x ≤ ρ and x > − d/c , or there are two negative roots: x ≤ ρ and x > − d/c and two positive roots: x , < ρ . (iii) If 0 ≤ − d < cρ , then there is one negative root smaller than ρ together with either one or three positive roots greater than orequal to − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ andone positive root x such that ρ ≤ x < − d/c . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ < − δ > − δ < cρ < < − δ < − δ < cρ .The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x ≥ max { µ , − d/c } . (iii) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (1)2 < x ≤ λ and a non-negative root x ≤ − d/c . (iv) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (2)2 < x ≤ ξ (1)2 and three positive roots: x such thatmax {− d/c, ξ (1)1 } ≤ x < µ , x such that µ < x ≤ µ and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo negative roots x , > ρ . (ii) If cρ ≤ − d < x ≤ ρ and another negative root x > − d/c . (iii) If 0 ≤ − d < cρ , then there is one negative root smaller than ρ together with either one or three positive roots greater than − d/c and smaller than or equal to ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ and apositive root x such that ρ ≤ x < − d/c . igure 1.7 Figure 1.8 b < , a < , c > , b < , a > , c < ,c < γ < < c < γ < c < c c < c < γ < c < < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ < ρ (pictured) or µ ≥ ρ .Obviously, − δ < cρ < < cρ .The minimum of x + ax + bx at negative x is greater thanthe minimum at positive x by the amount of − (1 / a (9 a − b ) / > − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d , then there is a negative root x ≤ λ and apositive root x such that min { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo negative roots (these are smaller than ρ if µ < ρ or greaterthan or equal to ρ if µ ≥ ρ ). (ii) If cρ ≤ − d < x ≤ ρ and another negative root x ≥ − d/c . (iii) If 0 ≤ − d < cρ , then there is one negative root x < ρ anda non-negative root x such that − d/c ≤ x < ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ and apositive root x such that ρ ≤ x ≤ − d/c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ > ρ (pictured) or µ ≤ ρ .Obviously, − δ < cρ < < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a positive root x ≤ µ andanother positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a negative root x suchthat min { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots (these are greater than ρ if µ > ρ or smallerthan or equal to ρ if µ ≤ ρ ). (ii) If cρ ≤ − d <
0, then there is a positive root x < − d/c andanother positive root x ≥ ρ . (iii) If 0 ≤ − d < cρ (pictured), then there is one non-positiveroot x such that ρ < x ≤ − d/c and a positive root x > ρ . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c ≤ x ≤ ρ and a positive root x > ρ . igure 1.9 Figure 1.10 b < , a > , c < , b < , a > , c < ,c < c < γ < c < < γ < c c < γ < c < c < < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ < − δ > − δ < cρ < < − δ < − δ < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x ≤ min {− d/c, µ } and another positive root x such that µ ≤ x <λ . (iii) If 0 ≤ − d < − δ , then there is a positive root x such that λ ≤ x < ξ (3)1 , and a non-positive root x such that ξ (3)2 < x ≤− d/c . (iv) If − δ ≤ − d < − δ , then there is a positive root x suchthat ξ (3)1 ≤ x < ξ (2)1 , a negative root x such that µ < x ≤ min { ξ (3)2 , − d/c } , another negative root x such that µ ≤ x <µ , and a third negative root x such that ξ (2)2 < x ≤ µ . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots x , < ρ . (ii) If cρ ≤ − d < x < − d/c and another positive root x ≥ ρ . (iii) If 0 ≤ − d < cρ , then there is one positive root x > ρ together with either three negative roots x , , such that ρ
0. Also: − δ < − δ > − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ . Thus one can have − δ < cρ < − δ < < − δ < cρ (pictured) or − δ < − δ ≤ cρ < < − δ < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is a positive root x such that ξ (3)2 < x ≤ min {− d/c, µ } and another positive root x suchthat µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d < x such that λ < x ≤ µ , another negative root x such that µ ≤ x < λ , a positive root x < min { ξ (3)2 , − d/c } and anotherpositive root x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a negative root x such that λ ≤ x < µ , anon-positive root x such that µ < x ≤ − d/c and a positiveroot x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or thereare two positive roots x , < ρ , or there are two positive roots x , < ρ and two negative roots x , > ρ (the latter appearwhen cρ ≥ − δ ). (ii) If cρ ≤ − d < x , such that x < − d/c and x ≥ ρ or there are twopositive roots x , such that x < − d/c and x ≥ ρ and twonegative roots x , > ρ . (iii) If 0 ≤ − d < cρ , then there is one positive root x > ρ together with either one or three negative roots greater than ρ and smaller than or equal to − d/c . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and one positive root x > ρ . igure 1.11 Figure 1.12 b < , a > , c < , b < , a > , c > ,c < γ < c < c < < γ < c c < γ < c < < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ .Obviously, − δ < − δ < cρ < < − δ < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (1)2 < x ≤ µ and another negative root x such that µ ≤ x < ξ (1)1 . (iii) If − δ ≤ − d < x such that λ < x ≤ ξ (1)2 , another negative root x such that ξ (1)1 ≤ x < λ , a positive root x ≤ min {− d/c, µ } and anotherpositive root x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a negative root x such that λ ≤ x < µ , anon-positive root x such that µ < x ≤ − d/c and a positiveroot x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or there aretwo negative roots x , > ρ , or there are two negative roots x , > ρ together with two positive roots x , < ρ . (ii) If cρ ≤ − d < x , > ρ , a positive root x < − d/c and another positive root x ≥ ρ . (iii) If 0 ≤ − d < cρ , then there is one positive root bigger than ρ and either one or three negative roots greater than ρ andsmaller than or equal to − d/c . (iv) If cρ ≤ − d , then there is a negative root x such that − d/c < x ≤ ρ and a positive root x > ρ . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − δ < cρ < − δ < < − δ < cρ (pictured) or − δ < − δ ≤ cρ < < − δ < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is a negative root x such that ξ (1)2 < x ≤ µ and another negative root x suchthat max {− d/c, µ } ≤ x < ξ (1)1 . (iii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ ξ (1)2 , another negative root x ≥ max {− d/c, ξ (1)1 } , apositive root x such that λ < x ≤ µ and another positiveroot x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a non-negative root x such that − d/c ≤ x < µ ,a positive root x such that µ < x ≤ λ and another positiveroot x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { min { ρ , − d/c } , ξ (2)1 } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots, or there aretwo negative roots x , > ρ , or there are two negative roots x , > ρ together with two positive roots x , < ρ (the latterappear when cρ ≥ − δ ). (ii) If cρ ≤ − d < x , such that x ≤ ρ and x > − d/c , or there are twonegative roots x , such that x ≤ ρ and x > − d/c , togetherwith two positive roots x , < ρ . (iii) If 0 ≤ − d < cρ , then there is one negative root smaller than ρ together with either one or three positive roots greater than orequal to − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is a negative root x < ρ and apositive root x such that ρ ≤ x < − d/c . igure 1.13 Figure 1.14 b < , a > , c > , b < , a < , c > ,c < γ < c < < γ < c < c c < γ < c < < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ < − δ > − δ < cρ < < − δ < − δ < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x ≥ max { µ , − d/c } . (iii) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (1)2 < x ≤ λ and a non-negative root x such that − d/c ≤ x < ξ (1)1 . (iv) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (2)2 < x ≤ ξ (1)2 and three positive roots: x such thatmax { ξ (1)1 , − d/c } ≤ x < µ , x such that µ < x ≤ µ and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo negative roots x , > ρ . (ii) If cρ ≤ − d < x ≤ ρ and another negative root x > − d/c . (iii) If 0 ≤ − d < cρ , then there is one negative root smaller than ρ and either one or three positive roots greater than or equal to − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ and apositive root x such that ρ ≤ x < − d/c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ < ρ (pictured) or µ ≥ ρ .Obviously, − δ ≤ cρ < < cρ .The minimum of x + ax + bx at negative x is smaller than theminimum at positive x by the amount of (1 / a (9 a − b ) / .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x ≥ max { µ , − d/c } . (iii) If 0 ≤ − d , then there is a negative root x ≤ λ and apositive root x such that min { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d < cρ (pictured), then there are either no real roots orthere are two negative roots (these are smaller than ρ if µ < ρ or greater than or equal to ρ if µ ≥ ρ ). (ii) If cρ ≤ − d <
0, then there is a negative root x ≤ ρ andanother negative root x > − d/c . (iii) If 0 ≤ − d < cρ , then there is one negative root x < ρ andone positive root x such that − d/c ≤ x < ρ . (iv) If cρ ≤ − d , then there is one negative root x < ρ andone positive root x such that ρ ≤ x < − d/c . igure 2.1 Figure 2.2 < b ≤ a , a < , c < , < b ≤ a , a < , c < ,c < c < γ < < c < γ < c c < c < γ < < c < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ < ρ (pictured) or µ ≥ ρ .Obviously, − δ ≤ cρ < cρ < − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that min { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , µ } andanother positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d , then there is a non-positive root x ≥ − d/c anda positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots (these are greater than ρ if µ > ρ or smallerthan ρ if µ ≤ ρ ). (ii) If cρ ≤ − d < cρ (pictured), then there is a positive root x such that ρ < x ≤ − d/c and another positive root x ≥ ρ . (iii) If cρ ≤ − d <
0, then there is a positive root x such that − d/c < x ≤ ρ and another positive root x > ρ . (iv) If 0 ≤ − d , then there is one non-positive root x ≥ − d/c and a positive root x > ρ . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ <
0. Also: − δ < − δ < − δ < cρ < cρ < − δ < − δ < − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is one pos-itive root x such that max { ξ (3)2 , min { ρ , − d/c }} < x ≤ min { max { ρ , − d/c } , µ } and another positive root x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d < − δ , then there are four positive roots: x such that max { ξ , − d/c } < x ≤ µ , x such that µ ≤ x <µ , x such that µ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x <ξ (2)1 . (iv) If − δ ≤ − d <
0, then there is a positive root x suchthat − d/c < x ≤ ξ (2)2 and another positive root x such that ξ (2)1 ≤ x < λ . (v) If 0 ≤ − d , then there is a non-positive root x ≥ − d/c and apositive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots x , < ρ . (ii) If cρ ≤ − d < cρ , then there is a positive root x such that ρ < x ≤ − d/c and another positive root x ≥ ρ . (iii) If cρ ≤ − d < x > ρ together with either one or three positive roots greaterthan − d/c and smaller than or equal to ρ . (iv) If 0 ≤ − d , then there is one non-positive root x ≥ − d/c and a positive root x > ρ . igure 2.3 Figure 2.4 < b ≤ a , a < , c < , < b ≤ a , a < , c > ,c < γ < c < < c < γ < c c < γ < < c < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ >
0. Also: − δ < − δ < − δ < cρ < cρ < − δ < < − δ .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is one pos-itive root x such that max { ξ (3)2 , min { ρ , − d/c }} < x ≤ min { max { ρ , − d/c } , µ } and another positive root x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d <
0, then there are four positive roots: x suchthat − d/c < x ≤ µ , x such that µ ≤ x < λ , x such that λ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a non-positive root x ≥ max { ξ (2)2 , − d/c } and three positive roots: x such that λ ≤ x <µ , x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo positive roots x , < ρ . (ii) If cρ ≤ − d < cρ , then there is a positive root x such that ρ < x ≤ − d/c and another positive root x ≥ ρ . (iii) If cρ ≤ − d < x , , such that − d/c < x , , ≤ ρ together witha positive root x > ρ , or there is one negative root x such that − d/c < x ≤ ρ and a positive root x > ρ . (iv) If 0 ≤ − d , then there is one non-positive root greater thanor equal to − d/c , a positive root greater than ρ and either zeroor two positive roots smaller than ρ . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ > − δ < − c x − d with c = (1 / a ( b − a / > d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As 0 < c < c , one has − δ < − δ .One could have − δ < − δ < < cρ < − δ < cρ (pictured) or − δ < − δ < < cρ < cρ ≤ − δ .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x suchthat ξ (3)2 < x ≤ µ and another positive root x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ µ , another negative root x such that µ ≤ x < − d/c , a positive root x such that λ < x ≤ ξ (3)2 and anotherpositive root x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ (pictured), then there is a negativeroot x such that ξ (2)2 < x ≤ λ , a non-negative root x ≤ min {− d/c, µ } , a positive root x such that µ < x ≤ λ andanother positive x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 anda positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there aretwo positive roots smaller than ρ , or there are two positive rootssmaller than ρ and two negative roots smaller than or equal to − d/c . (ii) If 0 ≤ − d < cρ , then there is one negative root x , a non-negative root x ≤ − d/c and two positive roots x , such that ρ < x , < ρ . (iii) If cρ ≤ − d < cρ (pictured), then there is one negativeroot, either zero or two positive roots smaller than or equal to ρ ,and one positive root greater than − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is one negative root x and apositive root x such that ρ ≤ x < − d/c and either zero ortwo positive roots smaller than or equal to ρ (the latter appearwhen cρ ≤ − δ ). igure 2.5 Figure 2.6 < b ≤ a , a < , c > , < b ≤ a , a < , c > ,c < γ < < c < c < γ < c c < γ < < c < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ > − δ < − c x − d with c = (1 / a ( b − a / > d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c > c , one has − δ < − δ .Obviously, − δ < − δ < < cρ < − δ < cρ .The graph of x + ax + bx is shown on its own in the top-rightcorner.Point µ could be in the first quadrant or in the fourth (pictured).Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is a negative root x such that ξ (1)2 < x ≤ µ and another negative root x such that µ ≤ x < min {− d/c, ξ (1)1 } . (iii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ ξ (1)2 , another negative root x such that ξ (1)1 ≤ x < − d/c, a positive root x such that λ < x ≤ µ and anotherpositive root x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a non-negative root x ≤ − d/c , a positive root x such that µ < x ≤ λ , and another positive root x suchthat λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x ≤ ξ (2)2 and a positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there aretwo negative roots smaller than − d/c , or there are two negativeroots smaller than − d/c and two positive roots greater than ρ and smaller than ρ . (ii) If 0 ≤ − d < cρ , then there is one negative root x , a non-negative root x ≤ − d/c and two positive roots greater than ρ and smaller than ρ . (iii) If cρ ≤ − d < cρ (pictured), then there is one negativeroot and either one or three positive roots greater than or equalto − d/c and smaller than ρ . (iv) If cρ ≤ − d , then there is one negative root x and a positiveroot x such that ρ ≤ x < − d/c . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ > − δ < − δ < < − δ < cρ < − δ < cρ .Point µ could be in the first quadrant (pictured) or in the fourth.Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (1)2 < x ≤ λ and a non-negative root x < min {− d/c, ξ (1)1 } . (iv) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (2)2 < x ≤ ξ (1)2 and three positive roots: x such that ξ (1)1 ≤ x < µ , x such that µ < x ≤ µ and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x ≤ ξ (2)2 and a positive root x such that max { ξ (2)1 , min { ρ , − d/c } ≤ x < max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cρ , then there is one negative root, a non-negative root smaller than or equal to − d/c and either zero ortwo positive roots greater than ρ and smaller than ρ . (iii) If cρ ≤ − d < cρ (pictured), then there is one negativeroot, either zero or two positive roots smaller than or equal to ρ ,and one positive root greater than or equal to − d/c and smallerthan ρ . (iv) If cρ ≤ − d , then there is one negative root x and apositive root x such that ρ ≤ x < − d/c . igure 2.7 Figure 2.8 < b ≤ a , a < , c > , < b ≤ a , a > , c < ,c < γ < < c < γ < c < c c < c < γ < c < < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < < cρ < cρ . Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d < cρ (pictured), then there is a negative root x ≤ λ and a non-negative root x ≤ − d/c . (iv) If cρ ≤ − d , then there is a negative root x < λ and apositive root x such that min { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cρ (pictured), then there is a negative root x and a non-negative root x ≤ − d/c . (iii) If cρ ≤ − d < cρ , then there is one negative root x and apositive root x such that − d/c ≤ x < ρ . (iv) If cρ ≤ − d , then there is one negative root x and apositive root x such that ρ ≤ x < − d/c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < < cρ < cρ . Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a positive root x such that − d/c < x ≤ µ and another positive root x such that µ ≤ x <λ . (iii) If 0 ≤ − d < cρ (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . (iv) If cρ ≤ − d , then there is a positive root x > λ and anegative root x such that min { ρ , − d/c } ≤ x ≤ max { ρ , − d/c } . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cρ (pictured), then there is a positive root x and a non-positive root x ≥ − d/c . (iii) If cρ ≤ − d < cρ , then there is one positive root x and anegative root x such that ρ < x ≤ − d/c . (iv) If cρ ≤ − d , then there is one positive root x and anegative root x such that − d/c < x ≤ ρ . igure 2.9 Figure 2.10 < b ≤ a , a > , c < , < b ≤ a , a > , c < ,c < c < γ < c < < γ < c c < γ < c < c < < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ >
0. Also: − δ < − δ > − δ < < − δ < cρ < − δ < cρ .Point µ could be in the second quadrant (pictured) or in thethird.Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a positive root x such that − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d < − δ (pictured), then there is a negative root x such that x ≥ max { ξ (3)2 , − d/c } , and a positive root x such that λ ≤ x < ξ (3)1 . (iv) If − δ ≤ − d < − δ , then there is a negative root x suchthat ξ (2)2 < x ≤ min {− d/c, µ } , a negative root x such thatmax {− d/c, µ } ≤ x < µ , another negative root x such that µ < x ≤ ξ (3)2 , and a positive root x such that ξ (3)1 ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such thatmin { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } and a positiveroot x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater that − d/c . (ii) If 0 ≤ − d < cρ (pictured), then there is one positive root,one non-positive root greater than or equal to − d/c and eitherzero or two negative roots greater than ρ and smaller than orequal to ρ . (iii) If cρ ≤ − d < cρ , then there is one positive root, onenegative root greater than ρ and smaller than or equal to − d/c ,and either zero or two negative roots smaller than or equal to ρ . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and one positive root x . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ . Obviously, − δ < − δ <
0, then there is a negative root x such that λ < x ≤ µ , another negative root x such that µ ≤ x < λ , apositive root x such that − d/c < x ≤ ξ (3)2 and another positiveroot x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a negative root x such that λ ≤ x < µ ,a non-positive root x ≥ max { µ , − d/c } and a positive root x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x such that min { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } anda positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there aretwo positive roots greater than − d/c , or there are two positiveroots greater than − d/c and two negative roots greater than ρ and smaller than ρ . (ii) If 0 ≤ − d < cρ , then there is one positive root, one non-positive root greater than or equal to − d/c and two negative rootsgreater than ρ and smaller than ρ . (iii) If cρ ≤ − d < cρ (pictured), then there is one positive roottogether with either one or three negative roots greater than ρ and smaller than or equal to − d/c . (iv) If cρ ≤ − d , then there is one negative root x such that − d/c < x ≤ ρ and one positive root x . igure 2.11 Figure 2.12 < b ≤ a , a > , c < , < b ≤ a , a > , c > ,c < γ < c < c < < γ < c c < γ < c < < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ <
0. Also: − δ < − δ > − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ . One can have − δ < − δ <
0, then there is a negative root x such that λ < x ≤ ξ (1)2 , another negative root x such that ξ (1)1 ≤ x <λ , a positive root x such that − d/c < x ≤ µ and anotherpositive root x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a negative root x such that λ ≤ x < µ ,a non-positive root x ≥ max { µ , − d/c } and a positive root x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x such that min { ρ , − d/c } ≤ x ≤ min { max { ρ , − d/c } , ξ (2)2 } anda positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there are twonegative roots greater than ρ and smaller than ρ , or there aretwo negative roots greater than ρ and smaller than ρ and twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cρ , then there is one positive root, one non-positive root greater than or equal to − d/c and two negative rootsgreater than ρ and smaller than ρ . (iii) If cρ ≤ − d < cρ (pictured), then there is one positive root,a negative root greater than ρ and smaller than or equal to − d/c and either zero or two negative roots smaller than or equal to ρ . (iv) If cρ ≤ − d , then there is one negative root x suchthat − d/c < x ≤ ρ , one positive root x and either zeroor two negative roots greater than ρ (the latter appear when cρ ≤ − δ ). Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ >
0. Also: − δ < − δ < − δ < cρ < cρ < − δ < < − δ .Consideration of whether − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is a negative root x such that ξ (1)2 < x ≤ µ and another negative root x suchthat max { min { ρ , − d/c } , µ } ≤ x < min { ξ (1)1 , max { ρ , − d/c }} . (iii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ ξ (1)2 , a negative root x such that ξ (1)1 ≤ x < λ ,another negative root x such that λ < x ≤ µ and a fourthnegative root x such that µ ≤ x < − d/c . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , a negative root x such that λ ≤ x < µ , athird negative root x such that µ < x ≤ λ and a non-negativeroot x ≤ min {− d/c, ξ (2)1 } . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and apositive root x such that ξ (2)1 ≤ x < − d/c . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo negative roots x , such that ρ < x , < ρ . (ii) If cρ ≤ − d < cρ , then there are two negative roots x , such that x ≤ ρ and − d/c ≤ x < ρ . (iii) If cρ ≤ − d < ρ and either one or three negative roots greaterthan or equal to ρ and smaller than − d/c . (iv) If 0 ≤ − d , then there is a negative root smaller than ρ , either zero or two negative roots greater than ρ and onenon-negative root smaller than or equal to − d/c . igure 2.13 Figure 2.14 < b ≤ a , a > , c > , < b ≤ a , a > , c > ,c < γ < c < < γ < c < c c < γ < c < < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ < . Also: − δ < − δ < − δ < cρ < cρ < − δ < − δ < − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is a negative root x such that ξ (1)2 < x ≤ µ and another negative root x suchthat max { min { ρ , − d/c } , µ } ≤ x < min { ξ (1)1 , max { ρ , − d/c }} . (iii) If − δ ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ ξ (1)2 , a negative root x such that ξ (1)1 ≤ x < µ ,another negative root x such that µ < x ≤ µ and a fourthnegative root x such that µ ≤ x < min {− d/c, ξ (2)1 } . (iv) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ ξ (2)2 and a negative root x such that ξ (2)1 ≤ x < − d/c . (v) If 0 ≤ − d , then there is a negative root x ≤ λ and a non-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there aretwo negative roots x , such that ρ < x , < ρ . (ii) If cρ ≤ − d < cρ , then there are two negative roots x , such that x ≤ ρ and − d/c < x < ρ . (iii) If cρ ≤ − d < ρ and either one or three negative roots greaterthan or equal to ρ and smaller than − d/c . (iv) If 0 ≤ − d , then there is a negative root smaller than ρ andone non-negative root smaller than or equal to − d/c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).One could have µ > ρ (pictured) or µ ≤ ρ .Obviously, − δ ≤ cρ < cρ < − d < cρ or − d > cρ would yieldsharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x such thatmax { min { ρ , − d/c } , µ } ≤ x < max { ρ , − d/c } . (iii) If 0 ≤ − d , then there is a negative root x ≤ λ and anon-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d < cρ , then there are either no real roots or there are twonegative roots (these are smaller than ρ if µ < ρ and greaterthan ρ if µ ≥ ρ . (ii) If cρ ≤ − d < cρ (pictured), then there are two negativeroots x , such that x ≤ ρ and − d/c ≤ x < ρ . (iii) If cρ ≤ − d <
0, then there is a negative root smaller than ρ and another negative root greater than or equal to ρ and smallerthan − d/c . (iv) If 0 ≤ − d , then there is a negative root smaller than ρ andnon-negative root smaller than or equal to − d/c . igure 3.1 Figure 3.2 a < b ≤ a , a < , c < , a < b ≤ a , a < , c < ,c < c < γ < c < γ < < c c < c < γ < c < γ < < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < cσ h + h < cσ H + H < − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that − d/c < x ≤ µ and another positive root x suchthat µ ≤ x < λ . (iii) If 0 ≤ − d , then there is a non-positive root x ≥ − d/c anda positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cσ h + h , then there are either no real roots or thereare two positive roots x , > σ h . (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there aretwo positive roots: x such that max { σ H , − ( d + h ) /c } < x < min { σ h , − ( d + H ) /c } and x > σ h . (iii) If cσ H + H ≤ − d <
0, then there are two positive roots: x such that − d/c < x ≤ σ H and x > σ h . (iv) If 0 ≤ − d , then there is one non-positive root x ≥ − d/c and one positive root x > σ h . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ < − δ <
0. Also: − δ < − δ < cσ h + h < cσ H + H < − δ < − δ < − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is one positive root x such that max { ξ (3)2 , − d/c } < x ≤ µ and another positiveroot x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d < − δ , then there are four positive roots: x such that max { ξ (2)2 , − d/c } < x ≤ µ , x such that µ ≤ x <µ , x such that µ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x <ξ (2)1 . (iv) If − δ ≤ − d <
0, then there is a positive root x suchthat − d/c < x ≤ ξ (2)2 and another positive root x such that ξ (2)1 ≤ x < λ . (v) If 0 ≤ − d , then there is a non-positive root x ≥ − d/c and apositive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cσ h + h , then there are either no real roots or thereare two positive roots x , > σ h . (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is a positiveroot x such that max { σ H , − ( d + h ) /c } < x ≤ min { σ h , − ( d + H ) /c } and a positive root x > σ h . (iii) If cσ H + H ≤ − d <
0, then there is a positive root greaterthan σ h and either one or three positive roots greater than − d/c and smaller than or equal to σ H . (iv) If 0 ≤ − d , then there is one non-positive root x ≥ − d/c and one positive root x > σ h . igure 3.3(continues on next page) a < b ≤ a , a < , c < ,c < γ < c < c < γ < < c Notes (apply to all panes) As c < c < c , the quartic has three stationary points µ i and thenumber of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i (only shown onthe top-left pane).As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ > − δ <
0. Also: − δ < − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ .One can have cσ h + h < < cσ H + H (top-left pane), cσ h + h
0, then there are four positive roots: x such that − d/c < x ≤ µ , x such that µ ≤ x < λ , x such that λ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is one non-positive root x ≥ max { ξ (2)2 , − d/c } and tree positive roots: x such that λ ≤ x < µ ,x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (continues on next page)
Top-left pane
One can have either − δ > cσ H + H > > − δ > cσ h + h > − δ (pictured) or − δ > cσ H + H > > cσ h + h ≥ − δ > − δ . (i) If − d < cσ h + h , then there are either no real roots, or there are two positive roots greater than σ h , or there are two positive rootsgreater than σ h and another pair of positive roots greater than − d/c and smaller than σ H (the latter appear when cσ h + h ≥ − δ ). (ii) If cσ h + h ≤ − d < x such that max { σ H , − ( d + h ) /c } < x ≤ min { σ h , − ( d + H ) /c } ,another positive root x > σ h and either zero or two positive roots x , greater than − d/c and smaller than σ H ( x , are alwayspresent if cσ h + h ≥ − δ , while for the pictured cσ h + h < − δ the roots x , may or may not be there). (iii) If 0 ≤ − d < cσ H + H , then there is one non-positive root x ≥ − d/c , a positive root x < σ H , another positive root x suchthat σ H < x < − ( d + H ) /c , and a third positive root x > σ h . (iv) If cσ H + H ≤ − d , then there is one negative root x > − d/c , a positive root x > σ h and either zero or two positive rootssmaller than or equal to σ H . igure 3.3(continued from previous page) a < b ≤ a , a < , c < ,c < γ < c < c < γ < < c Analysis based on solving quadratic equations only — continued from previous page
Top-right pane
One can have either − δ > > − δ > cσ H + H > cσ h + h > − δ , or − δ > > cσ H + H ≥ − δ > cσ h + h > − δ , or − δ > > cσ H + H > cσ h + h ≥ − δ > − δ (when σ h and σ H are close). (i) If − d < cσ h + h , then there are either no real roots, or there are two positive roots greater than σ h , or there are two positive rootsgreater than σ h and two positive roots greater than − d/c and smaller than σ H (the latter appear when cσ h + h ≥ − δ ). (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is a positive root x such that max { σ H , − ( d + h ) /c } < x ≤ min { σ h , − ( d + H ) /c } and a positive root x > σ h . If − δ > cσ H + H , there are no other roots. If cσ h + h ≥ − δ , there are two more positive roots x , greater than − d/c and smaller than σ H . If cσ H + H ≥ − δ > cσ h + h , the roots x , may or may not be there. (iii) If cσ H + H ≤ − d <
0, then there is a positive root greater than σ h and, if − δ > cσ H + H , there is either one or three positiveroots greater than − d/c and smaller than σ H . If however, cσ H + H ≥ − δ , then there will be a positive root greater than σ h andthree positive roots greater than − d/c and smaller than σ H . (iv) If 0 ≤ − d , then there is a non-positive root greater than or equal to − d/c , either zero or two positive roots smaller than σ H ,and a positive root greater than σ h . Bottom-right pane
One can only have − δ > cσ H + H > cσ h + h > > − δ > − δ . (i) If − d <
0, then there are either no real roots, or there are two positive roots greater than σ h , or there are four positive roots: twogreater than σ h and the other two — greater than − d/c and smaller than σ H . (ii) If 0 ≤ − d < cσ h + h , then there is one non-positive root greater than − d/c , one positive root smaller than σ H , and two positiveroots greater than σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is one negative root x > − d/c , a positive root x < σ H , a positive root x such that max { σ H , − ( d + h ) /c } < x ≤ min {− ( d + H ) /c, σ h } and another positive root x ≥ σ h . (iv) If cσ H + H ≤ − d , then there is a negative root greater than − d/c , a positive root greater than σ h and either zero or two positiveroots smaller than σ H . igure 3.4 a < b ≤ a , a < , c < ,c < γ < c < c < γ < < c Notes As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx at three points: λ , , . Thus − δ > − δ <
0. Also: − δ < − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at two points: α and β = − α − a/
2, where α and β are the simultaneous double roots of the varied quartic x + ax + bx + c x + d .As c > c , one has − δ > − δ .Obviously, − δ < − δ < < cσ h + h < cσ H + H < − δ .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and also whether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x such that max { ξ (1)2 , − d/c } < x ≤ µ and another positive root x suchthat µ ≤ x < ξ (1)1 . (iii) If − δ ≤ − d <
0, then there are four positive roots: x such that − d/c < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x < λ , x such that λ < x ≤ µ , and x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ (pictured), then there is one non-positive root x ≥ max { ξ (2)2 , − d/c } and tree positive roots: x such that λ ≤ x < µ , x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there are two positive roots greater than − d/c and smaller than σ H , or there arefour positive roots, two of which greater than − d/c and smaller than σ H and the other two — greater than σ h . (ii) If 0 ≤ − d < cσ h + h , then there is one non-positive root x ≥ − d/c , a non-negative root x < σ H and twho positive roots x , > σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is one negative root x > − d/c , a positive root x < σ H , another positiveroot x such that max { σ H , − ( d + h ) /c } < x ≤ min { σ h , − ( d + H ) /c } , and a third positive root x > σ h . (iv) If cσ H + H ≤ − d , then there is a negative root x > − d/c and a positive root x > σ h and either zero or two positive rootssmaller than σ H . igure 3.6(Figure 3.5 is on next page) a < b ≤ a , a < , c > ,c < γ < c < γ < < c < c Notes (apply to both panes) As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx at one point only ( λ ). Thus − δ > − δ >
0. Also, − δ < − ˆ c = (1 / a [(9 / a / − b ] ≤ ≤ ˆ c < c .With the increase of c from zero towards c , one has: − δ < < − δ < cσ h + h < cσ H + H < − δ (pictured on left pane;0 < c < ˆ c < c ), then − δ < < − δ ≤ cσ H + H ≤ cσ h + h ≤ − δ (at c = ˆ c there is a swap between cσ H + H and cσ h + h ), then − δ < < cσ H + H ≤ − δ < cσ h + h ≤ − δ , or − δ < < − δ ≤ cσ H + H < − δ ≤ cσ h + h , and finally: − δ < < cσ H + H ≤− δ < − δ ≤ cσ h + h (pictured on right pane; 0 < ˆ c < c < c ).Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and also whether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (applies to both panes) (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a negative root x such that λ < x ≤ µ and a negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d < − δ (pictured), then there is a negative root x such that ξ (1)2 < x ≤ λ and a non-negative root x ≤ min { ξ (1)1 , − d/c } . (iv) If − δ ≤ − d < − δ , then there is one negative root x such that ξ (2)2 < x ≤ ξ (1)2 and tree positive roots: x such that ξ (1)1 ≤ x < µ , x such that µ < x ≤ µ , and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only < c < ˆ c < c (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cσ h + h (pictured), then there is a non-positiveroot x , a non-negative root x such that max { , − ( d + H ) /c } ≤ x < min {− d/c, σ H } and either zero or two positive roots greaterthan σ H and smaller than σ h . (iii) If cσ h + h ≤ − d < cσ H + H , then there is one negative root x , a positive root x such that − ( d + H ) /c < x < σ H , anotherpositive root x such that σ H < x ≤ σ h , and a third positiveroot x such that σ h ≤ x < − ( d + h ) /c . (iv) If cσ H + H ≤ − d , then there is a negative root x , a positiveroot x such that σ h ≤ x < − ( d + h ) /c and either zero or twopositive roots greater than or equal to − ( d + H ) /c and smallerthan σ h . Analysis based on solving quadratic equations only < ˆ c < c < c (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cσ H + H (pictured), then there is a non-positiveroot x , a non-negative root x such that max { , − ( d + H ) /c } ≤ x ≤ min {− d/c, σ H } , and either zero or two positive roots greaterthan σ H and smaller than σ h (the latter appear when − δ ≤ cσ H + H ). (iii) If cσ H + H ≤ − d < cσ h + h , then, if − δ ≤ cσ H + H and − δ ≥ cσ h + h , there will be one negative root together withthree positive roots greater than or equal to σ H and smaller than σ h . If however, cσ H + H < − δ or − δ < cσ h + h , there will beone negative root together with either one or three positive rootsgreater than or equal to σ H and smaller than σ h . (iv) If cσ h + h ≤ − d , then there is a negative root x , a positiveroot x such that σ h ≤ x < − ( d + h ) /c and either zero or twopositive roots greater than or equal to − ( d + H ) /c and smallerthan σ h (the latter appear when cσ h + h ≤ − δ ). igure 3.5 Figure 3.7 a < b ≤ a , a < , c < , a < b ≤ a , a < , c > ,c < γ < c < γ < c < < c c < γ < c < γ < < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ > − δ >
0. Also: − δ < − δ < < − δ < cσ h + h < cσ H + H < − δ .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d < − δ , then there is one non-positive root x > max {− d/c, ξ (1)2 } and one positive root x such that λ ≤ x <ξ (1)1 . (iv) If − δ ≤ − d < − δ (pictured), then there is one negativeroot x such that max { ξ (2)2 , − d/c } ≤ x < ξ (1)2 and tree positiveroots: x such that ξ (1)1 ≤ x < µ , x such that µ < x ≤ µ ,and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c and smaller than σ H . (ii) If 0 ≤ − d < cσ h + h , then there is one non-positive root x ≥ − d/c , one positive root smaller than σ H , and either zero ortwo positive roots greater than σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there isone negative root x > − d/c , a positive root x < σ H , an-other positive root x such that max { σ H , − ( d + h ) /c } < x ≤ min { σ h , − ( d + H ) /c } , and a third positive root x > σ h . (iv) If cσ H + H ≤ − d , then there is one negative root x > − d/c ,one positive root x > σ h and either zero or two positive rootssmaller than or equal to σ H . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < < cσ H + H < cσ h + h .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d (pictured), then there is a negative root x ≤ λ and a non-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cσ H + H (pictured), then there is a non-positiveroot x and a non-negative root x such that max { , − ( d + H ) /c } ≤ x ≤ min {− d/c, σ H } . (iii) If cσ H + H ≤ − d < cσ h + h , then there is one negative root x and one positive root x such that σ H ≤ x < σ h . (iv) If cσ h + h ≤ − d , then there is a negative root x and apositive root x such that σ h ≤ x < − ( d + h ) /c . igure 3.9(Figure 3.8 is on next page) a < b ≤ a , a > , c < ,c < c < < γ < c < γ < c Notes (apply to both panes) As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx at one point only ( λ ). Thus − δ > − δ >
0. Also, − δ < − ˆ c = (1 / a [(9 / a / − b ] ≥ c < ˆ c ≤ c from c towards zero, one has: − δ < < cσ H + H ≤ − δ < − δ ≤ cσ h + h (pictured on left pane; c < c < ˆ c < − δ < < cσ H + H ≤ − δ < cσ h + h ≤ − δ , or − δ < < − δ ≤ cσ H + H < − δ ≤ cσ h + h ,then − δ < < − δ ≤ cσ H + H ≤ cσ h + h ≤ − δ (at c = ˆ c there is a swap between cσ H + H and cσ h + h ), and finally − δ < < − δ < cσ h + h < cσ H + H < − δ (pictured on right pane; c < ˆ c < c < − d ≤ cσ h + h or − d > cσ h + h and also whether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (applies to both panes) (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is a positive root x such that − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d < − δ (pictured), then there is a non-positive root x ≥ max { ξ (3)2 , − d/c } and a positive root x such that λ ≤ x <ξ (3)1 . (iv) If − δ ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ µ , another negative root x such that µ ≤ x < µ , a third negative root x such that µ < x ≤ ξ (3)2 , and a positive root x such that ξ (3)1 ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only c < c < ˆ c < (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cσ H + H (pictured), then there is a non-negativeroot x , a non-positive root x such that max {− d/c, σ H } < x ≤ min { , − ( d + H ) /c } , and either zero or two positive roots greaterthan σ h and smaller than σ H (the latter appear when − δ ≤ cσ H + H ). (iii) If cσ H + H ≤ − d < cσ h + h , then, if cσ H + H < − δ or cσ h + h > − δ , there will be one positive root together with eitherone or three negative roots greater than σ h and smaller than orequal to σ H . If however, − δ ≤ cσ H + H and cσ h + h ≤ − δ ,there will be one positive root together with three negative rootsgreater than σ h and smaller than or equal to σ H . (iv) If cσ h + h ≤ − d , then there is one negative root x suchthat − ( d + h ) /c < x ≤ σ h , one positive root x and either zeroor two positive roots smaller than or equal to − ( d + H ) /c andgreater than σ h (the latter appear when cσ h + h < − δ ). Analysis based on solving quadratic equations only c < ˆ c < c < (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cσ h + h (pictured), then there is a non-negativeroot x , a non-positive root x such that max {− d/c, σ h } < x ≤ min { , − ( d + H ) /c } and either zero or two positive roots greaterthan or equal to σ h and smaller than σ H . (iii) If cσ h + h ≤ − d < cσ H + H , then there is one positive root x , a negative root x such that σ H < x < − ( d + H ) /c , anothernegative root x such that σ h ≤ x < σ H , and a third negativeroot x such that − ( d + h ) /c < x ≤ σ h . (iv) If cσ H + H ≤ − d , then there is a positive root x , a negativeroot x such that − ( d + h ) /c < x ≤ σ h and either zero or twopositive roots smaller than or equal to σ H and greater than σ h . igure 3.8 Figure 3.10 a < b ≤ a , a > , c < , a < b ≤ a , a > , c > ,c < c < < γ < c < γ < c c < < c < γ < c < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < < cσ H + H < cσ h + h .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cσ H + H (pictured), then there is a non-negativeroot x and a non-positive root x such that max {− d/c, σ H }
0. Also: − δ < − δ < < − δ < cσ h + h < cσ H + H < − δ .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x suchthat − λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d < − δ , then there is one negative root x such that ξ (3)2 < x ≤ λ and one non-negative root x ≤ min {− d/c, ξ (3)1 } . (iv) If − δ ≤ − d < − δ (pictured), then there is one negative root x such that ξ (2)2 < x ≤ µ , another negative root x such that µ ≤ x < µ , a third negative root x such that µ < x ≤ ξ (3)2 ,and a positive root x such that ξ (3)1 ≤ x < min { ξ (2)1 , − d/c } . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and apositive root x such that ξ (2)1 ≤ x < − d/c. Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c and greater than σ H . (ii) If 0 ≤ − d < cσ h + h , then there is one non-negative root x ≤ − d/c , one non-positive root greater than σ H , and eitherzero or two negative roots smaller than σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there isone positive root x < − d/c , a negative root x > σ H , an-other negative root x such that max { σ h , − ( d + H ) /c } ≤ x < min { σ H , − ( d + h ) /c } , and a third negative root x ≤ σ h . (iv) If cσ H + H ≤ − d , then there is one positive root x < − d/c ,one negative root x < σ h and either zero or two negative rootsgreater than or equal to σ H . igure 3.11 a < b ≤ a , a > c > ,c < < γ < c < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i (only shown on the top-left pane).As γ < c < γ , the straight line − cx intersects x + ax + bx at three points: λ , , . Thus − δ > − δ <
0. Also: − δ < − c x − d with c = (1 / a ( b − a / > d = (1 / b − a / > x + ax + bx at two points: α and β = − α − a/
2, where α and β are the simultaneous double roots of the varied quartic x + ax + bx + c x + d .As c < c , one has − δ < − δ .Obviously, − δ < − δ < < cσ h + h < cσ H + H < − δ .Consideration of whether − d ≤ cσ h + h or − d > cσ h + h and also whether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one negative root x such that − ξ (3)2 < x ≤ µ and another negative root x such that µ ≤ x < min {− d/c, ξ (3)1 } . (iii) If − δ ≤ − d <
0, then there are four negative roots: x such that λ < x ≤ µ , x such that µ ≤ x < λ , x such that λ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x < − d/c. (iv) If 0 ≤ − d < − δ (pictured), then there are three negative roots: x such that ξ (2)2 < x ≤ λ , x such that λ ≤ x < µ , and x such that µ < x ≤ λ , together with the non-negative root x ≤ min {− d/c, ξ (2)1 } . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and a positive root x such that ξ (2)1 ≤ x < − d/c. Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there are two negative roots smaller than − d/c and greater than σ H , or thereare four negative roots — two smaller than − d/c and greater than σ H and the other two — smaller than σ h . (ii) If 0 ≤ − d < cσ h + h , then there is one non-negative root x ≤ − d/c , one non-positive root greater than σ H , and two negativeroots smaller than σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is one positive root x < − d/c , a negative root x > σ H , another negativeroot x such that max { σ h , − ( d + H ) /c } ≤ x < min { σ H , − ( d + h ) /c } , and a third negative root x ≤ σ h . (iv) If cσ H + H ≤ − d , then there is one positive root x < − d/c , one negative root x < σ h and either zero or two negative rootsgreater than or equal to σ H . igure 3.12(continues on next page) a < b ≤ a , a > , c > ,c < < γ < c < c < γ < c Notes (apply to all panes) As c < c < c , the quartic has three stationary points µ i and thenumber of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i (only shown onthe top-left pane).As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ > − δ <
0. Also: − δ < − c x − d with c = (1 / a ( b − a / < d = (1 / b − a / > x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d .As c > c , one has − δ < − δ .One can have cσ h + h < < cσ H + H (top-left pane), cσ h + h
0, then there are four negative roots: x such that λ < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x < λ , x such that λ < x ≤ µ , and x such that µ ≤ x < − d/c. (iv) If 0 ≤ − d < − δ , then there are tree negative roots: x such that ξ (2)2 < x ≤ λ , x such that λ ≤ x < µ , and x such that µ < x ≤ λ , together with the non-negative root x ≤ min { ξ (2)1 , − d/c } . (v) If − δ ≤ − d , then there is a negative root x ≤ ξ (2)2 and a positive root x such that ξ (2)1 ≤ x − d/c . Analysis based on solving quadratic equations only (continues on next page)
Top-left pane
One can have either − δ > cσ H + H > > − δ > cσ h + h > − δ (pictured) or − δ > cσ H + H > > cσ h + h ≥ − δ > − δ . (i) If − d < cσ h + h , then there are either no real roots, or there are two negative roots smaller than σ h , or there are two negative rootssmaller than σ h and another pair of negative roots smaller than − d/c and greater than σ H (the latter appear when cσ h + h ≥ − δ ). (ii) If cσ h + h ≤ − d < σ h , another negative root greater than or equal tomax { σ h , − ( d + H ) /c } and smaller than min { σ H , − ( d + h ) /c } , and either zero or two negative roots smaller than − d/c and greaterthan σ H (the latter are always present if cσ h + h ≥ − δ , while for the pictured cσ h + h < − δ they may or may not be there). (iii) If 0 ≤ − d < cσ H + H , then there is one non-negative root x ≤ − d/c , a negative root x > σ H , another negative root x suchthat − ( d + H ) /c < x < σ H , and a third negative root x < σ h . (iv) If cσ H + H ≤ − d , then there is one positive root x < − d/c , a negative root x < σ h and either zero or two negative rootsgreater than or equal to σ H . igure 3.12(continued from previous page) a < b ≤ a , a > , c > ,c < < γ < c < c < γ < c Analysis based on solving quadratic equations only — continued from previous page
Top-right pane
One can have either − δ > > − δ > cσ H + H > cσ h + h > − δ , or − δ > > cσ H + H ≥ − δ > cσ h + h > − δ , or − δ > > cσ H + H > cσ h + h ≥ − δ > − δ (when σ h and σ H are close). (i) If − d < cσ h + h , then there are either no real roots, or there are two negative roots smaller than σ h , or there are two negativeroots smaller than σ h and two negative roots smaller than − d/c and greater than σ H (the latter appear when cσ h + h ≥ − δ ). (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is a negative root x such that max { σ h , − ( d + H ) /c } ≤ x < min { σ H , − ( d + h ) /c } and a negative root x < σ h . If − δ > cσ H + H , there are no other roots. If cσ h + h ≥ − δ , there are two more negative roots smallerthan − d/c and greater than σ H . If cσ H + H ≥ − δ > cσ h + h , these two roots may or may not be there. (iii) If cσ H + H ≤ − d <
0, then there is a negative root smaller than σ h and, if − δ > cσ H + H , there is either one or three negativeroots smaller than − d/c and greater than σ H . If however, cσ H + H ≥ − δ , then there will be a negative root smaller than σ h andthree negative roots smaller than − d/c and greater than σ H . (iv) If 0 ≤ − d , then there is a non-negaive root smaller than or equal to − d/c , either zero or two negative roots greater than σ H ,and a negative root smaller than σ h . Bottom-right pane
One can only have − δ > cσ H + H > cσ h + h > > − δ > − δ . (i) If − d <
0, then there are either no real roots, or there are two negative roots smaller than σ h , or there are four negative roots:two smaller than σ h and two greater than σ H and smaller than − d/c . (ii) If 0 ≤ − d < cσ h + h , then there is one non-negative root smaller than − d/c , one negative root greater than σ H , and two negativeroots smaller than σ h . (iii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is one positive root x < − d/c , a negative root x > σ H , a negative root x such that max {− ( d + H ) /c, σ h } ≤ x < min { σ H , − ( d + h ) /c } and another negative root x ≤ σ h . (iv) If cσ H + H ≤ − d , then there is a positive root smaller than − d/c , a negative root smaller than σ h and either zero or twonegative roots greater than σ H . igure 3.13 Figure 3.14 a < b ≤ a , a > , c > , a < b ≤ a , a > , c > ,c < < γ < c < γ < c < c c < < γ < c < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ < − δ <
0. Also: − δ < − δ < cσ h + h < cσ H + H < − δ < − δ < − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ (pictured), then there is one negativeroot x such that ξ (1)2 < x ≤ µ and another negative root x such that µ ≤ x < min { ξ (1)1 , − d/c } . (iii) If − δ ≤ − d < − δ , then there are four negative roots: x such that ξ (2)2 < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x < µ , x suchthat µ < x ≤ µ , and x such that µ ≤ x < min { ξ (2)1 , − d/c } . (iv) If − δ ≤ − d <
0, then there is a negative root x suchthat λ < x ≤ ξ (2)2 and another negative root x such that ξ (2)1 ≤ x < − d/c . (v) If 0 ≤ − d , then there is a non-negative root x ≤ − d/c anda negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cσ h + h , then there are either no real roots or thereare two negative roots x , < σ h . (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is a negativeroot x such that max { σ h , − ( d + H ) /c } ≤ x < min { σ H , − ( d + h ) /c } and a negative root x < σ h . (iii) If cσ H + H ≤ − d <
0, then there is a negative root smallerthan σ h and either one or three negative roots smaller than − d/c and greater than or equal to σ H . (iv) If 0 ≤ − d , then there is one non-negative root x ≤ − d/c and one negative root x < σ h . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, − δ < cσ h + h < cσ H + H < − d ≤ cσ h + h or − d > cσ h + h and alsowhether − d ≤ cσ H + H or − d > cσ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that λ < x ≤ µ and another negative root x suchthat µ ≤ x < − d/c . (iii) If 0 ≤ − d , then there is a non-negative root x ≤ − d/c anda negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cσ h + h , then there are either no real roots or thereare two negative roots x , < σ h . (ii) If cσ h + h ≤ − d < cσ H + H (pictured), then there is a negativeroot x such that max { σ h , − ( d + H ) /c } ≤ x < min { σ H , − ( d + h ) /c } and another negative root x < σ h . (iii) If cσ H + H ≤ − d <
0, then there is a negative root smallerthan σ h and another negative root smaller than − d/c and greaterthan or equal to σ H . (iv) If 0 ≤ − d , then there is one non-negative root x ≤ − d/c and one negative root x < σ h . igure 4.1 Figure 4.2 a < b ≤ a , a < , c < , a < b ≤ a , a < , c < ,c < c < γ < c < γ < c < c < c < γ < c < γ < c < Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c <
0, one has − δ < cσ H + H < cσ h + h < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < x such that − d/c < x ≤ µ and another positive root x suchthat µ ≤ x < λ . (iii) If 0 ≤ − d , then there is a non-positive root x ≥ − d/c anda positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots or thereare two positive roots x , > − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h (pictured), then there are twopositive roots: x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and x > τ H . (iii) If cτ h + h ≤ − d <
0, then there are two positive roots: x such that − d/c < x ≤ − ( d + h ) /c and x > τ H . (iv) If 0 ≤ − d , then there is one non-positive root x ≥ − d/c and one positive root x > τ H . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ < − δ <
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c <
0, one has − δ < cσ H + H < − δ < cσ h + h < − δ < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x suchthat max { ξ (3)2 , − d/c } < x ≤ µ and another positive root x such that µ ≤ x < ξ (3)1 . (iii) If − δ ≤ − d < − δ , then there are four positive roots: x such that max { ξ (2)2 , − d/c } < x ≤ µ , x such that µ ≤ x <µ , x such that µ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x <ξ (2)1 . (iv) If − δ ≤ − d <
0, then there is a positive root x suchthat − d/c < x ≤ ξ (2)2 and another positive root x such that ξ (2)1 ≤ x < λ . (v) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots or thereare two positive roots x , > − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h , then there are two positiveroots: x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and x > τ H ,together with either zero or two positive roots x , such that − d/c < x , < τ h . (iii) If cτ h + h ≤ − d <
0, then there are two positive roots: x such that − d/c < x ≤ − ( d + h ) /c and x > τ H , together witheither zero or two positive roots greater than or equal to τ h andsmaller than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > τ H . igure 4.3(continues on next page) a < b ≤ a , a < , c < ,c < γ < c < c < γ < c < Notes (apply to all panes) As c < c < c , the quartic has three stationary points µ i and thenumber of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i (only shown onthe top-left pane).As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ > − δ <
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c , one has − δ < − δ and also cτ H + H < cτ h + h .Top-left pane: one can have − δ < cτ H + H < − δ < < cτ h + h < − δ (pictured) or − δ < − δ ≤ cτ H + H < < cτ h + h < − δ .Top-right pane ( c closer to γ ): one can have − δ < cτ H + H
0, then there are four positive roots: x such that − d/c < x ≤ µ , x such that µ ≤ x < λ , x such that λ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is one non-positive root x ≥ max { ξ (2)2 , − d/c } and tree positive roots: x such that λ ≤ x < µ ,x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (continues on next page)
Top-left pane(i) If − d < cτ H + H , then there are either no real roots, or there are two positive roots x , > − ( d + H ) /c , or there are two positiveroots x , > − ( d + H ) /c together with two positive roots x , such that − d/c < x , < τ h ( x , appear when cτ H + H ≥ − δ ). (ii) If cτ H + H ≤ − d <
0, then there is one positive root x such that − ( d + h ) /c < x ≤ − ( d + H ) /c , another positive root x > τ H and either zero or two positive roots x , such that − d/c < x , < τ h ( x , are always present if cτ H + H ≥ − δ , while for thepictured cτ H + H < − δ the roots x , may or may not be there). (iii) If 0 ≤ − d < cτ h + h , then there is one non-positive root x ≥ − d/c , a positive root x < τ h , another positive root x such that − ( d + h ) /c < x < − ( d + H ) /c , and a third positive root x > τ H . (iv) If cτ h + h ≤ − d (pictured), then there is one negative root x > − d/c , a positive root x > τ H and either zero or two positiveroots greater than τ h and smaller than or equal to − ( d + H ) /c . igure 4.3(continued from previous page) a < b ≤ a , a < , c < ,c < γ < c < c < γ < c < Analysis based on solving quadratic equations only — continued from previous page
Top-right pane(i) If − d < cτ H + H , then there are either no real roots, or there are two positive roots greater than − ( d + H ) /c , or there are twopositive roots greater than − ( d + H ) /c together with two positive roots greater than − d/c and smaller than τ h (the latter appearwhen cτ H + H ≥ − δ ). (ii) If cτ H + H ≤ − d < cτ h + h , then there is a positive root x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and a positive root x > τ H .If − δ > cτ h + h > cτ H + H , there are no other roots. If cτ h + h > cτ H + H ≥ − δ , there are two more positive roots x , greaterthan − d/c and smaller than τ h . If cτ h + h ≥ − δ > cτ H + H , the roots x , may or may not be there. (iii) If cτ h + h ≤ − d <
0, then, if − δ ≤ cσ h + h , there will be a positive root x > τ H , two positive roots x , such that τ h ≤ x , < − ( d + H ) /c , and a positive root x such that − d/c < x < σ h . If however, cσ h + h < − δ , then the roots x and x mayor may not be there. In summary: there is one positive root x > τ H and either one or three positive roots greater than − d/c andsmaller than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is a non-positive root greater than or equal to − d/c , either zero or two positive roots greaterthan τ h and smaller than − ( d + H ) /c , and a positive root greater than τ H . Bottom-right pane(i) If − d <
0, then there are either no real roots, or there are two positive roots greater than τ H , or there are four positive roots: twogreater than τ H and two greater than − d/c and smaller than τ h . (ii) If 0 ≤ − d < cτ H + H , then there is one non-positive root greater than − d/c , a positive root x < τ h , and two positive roots x , > − ( d + H ) /c . (iii) If cτ H + H ≤ − d < cτ h + h , then there is one negative root x > − d/c , a positive root x < τ h , a positive root x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and another positive root x > τ H . (iv) If cτ h + h ≤ − d (pictured), then there is a negative root greater than − d/c , a positive root greater than τ H and either zero ortwo positive roots greater than τ h and smaller than or equal to − ( d + H ) /c . igure 4.4 a < b ≤ a , a < , c < ,c < γ < c < c < γ < c < Notes As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx at three points: λ , , . Thus − δ < − δ >
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at two points: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c = c , one has − δ = − δ and cτ h + h = cτ H + H .As c < c <
0, obviously − δ < − δ < < cτ h + h < cτ H + H < − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and also whether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x such that max { ξ (1)2 , − d/c } < x ≤ µ and another positive root x suchthat µ ≤ x < ξ (1)1 . (iii) If − δ ≤ − d <
0, then there are four positive roots: x such that − d/c < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x < λ , x such that λ < x ≤ µ , and x such that µ ≤ x < λ . (iv) If 0 ≤ − d < − δ , then there is one non-positive root x ≥ max { ξ (2)2 , − d/c } and tree positive roots: x such that λ ≤ x < µ ,x such that µ < x ≤ λ , and x such that λ ≤ x < ξ (2)1 . (v) If − δ ≤ − d (pictured), then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there are two positive roots greater than − d/c and smaller than τ h , or there arefour positive roots, two of which greater than − d/c and smaller than τ h and the other two — greater than − ( d + H ) /c . (ii) If 0 ≤ − d < cτ h + h , then there is one non-positive root x ≥ − d/c , and three positive roots: x < τ h and x , > − ( d + H ) /c . (iii) If cτ h + h ≤ − d < cτ H + H , then there is one negative root greater than − d/c , a positive root greater than or equal to τ h andsmaller than τ H , and two positive roots greater than − ( d + H ) /c . (iv) If cτ H + H ≤ − d (pictured), then there is a negative root x > − d/c , a positive root x > τ H and either zero or two positiveroots greater than τ h and smaller than or equal to − ( d + H ) /c . igure 4.5 Figure 4.6 a < b ≤ a , a < , c < , a < b ≤ a , a < , c < ,c < γ < c < γ < c < c < c < γ < c < γ < c < c < Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ > − δ >
0. Also: − δ < − δ < < cτ h + h < − δ
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d < − δ (pictured), then there is one non-positiveroot x ≥ max {− d/c, ξ (1)2 } and one positive root x such that λ ≤ x < ξ (1)1 . (iv) If − δ ≤ − d < − δ , then there is one negative root x suchthat max { ξ (2)2 , − d/c } ≤ x < ξ (1)2 and tree positive roots: x such that ξ (1)1 ≤ x < µ , x such that µ < x ≤ µ , and x such that µ ≤ x < ξ (2)1 . (v) If − δ ≤ − d , then there is a negative root x such that − d/c < x ≤ ξ (2)2 and a positive root x ≥ ξ (2)1 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c and smaller than τ h . (ii) If 0 ≤ − d < cτ h + h , then there is one non-positive root x ≥ − d/c , one positive root smaller than τ h and either zero ortwo positive roots greater than − ( d + H ) /c (the latter appearwhen − δ ≤ cτ h + h ). (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onenegative root greater than − d/c , a positive root greater than orequal to τ h and smaller than τ H , and either zero or two positiveroots greater than − ( d + H ) /c (the latter are always present if − δ ≤ cτ h + h ). (iv) If cτ H + H ≤ − d , then there is a negative root x > − d/c ,a positive root x > τ H and either zero or two positive rootsgreater than τ h and smaller than or equal to − ( d + H ) /c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c and smaller than τ h . (ii) If 0 ≤ − d < cτ h + h , then there is one non-positive root x ≥ − d/c and one positive root smaller than τ h . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onenegative root greater than − d/c and one positive root greaterthan or equal to τ h and smaller than τ H . (iv) If cτ H + H ≤ − d , then there is one negative root x > − d/c and one positive root x ≥ τ H . igure 4.7 Figure 4.8 a < b ≤ a , a < , c > , a < b ≤ a , a > , c < ,c < γ < c < γ < c < < c c < < c < γ < c < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d (pictured), then there is a negative root x ≤ λ and a non-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cτ h + h , then there is a non-positive root x and a non-negative root x such that max { , − ( d + h ) /c } ≤ x < min {− d/c, τ h } . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onenegative root x and one positive root x such that max { τ h , − ( d + H ) /c } ≤ x < min {− ( d + h ) /c, τ H } . (iv) If cτ H + H ≤ − d , then there is a negative root x and apositive root x such that τ H ≤ x < − ( d + H ) /c . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cτ h + h , then there is a non-negative root x and a non-positive root x such that max {− d/c, τ h } < x ≤ min { , − ( d + h ) /c } . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onepositive root x and one negative root x such that max {− ( d + h ) /c, τ H } < x ≤ min { τ h , − ( d + H ) /c } . (iv) If cτ H + H ≤ − d , then there is a positive root x and anegative root x such that − ( d + H ) /c < x ≤ τ H . igure 4.9 Figure 4.10 a < b ≤ a , a > , c > , a < b ≤ a , a > , c > , < c < c < γ < c < γ < c < c < c < γ < c < γ < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x suchthat − λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d (pictured), then there is one negative root x ≤ λ and one non-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c and greater than τ h . (ii) If 0 ≤ − d < cτ h + h , then there is one non-negative root x ≤ − d/c and one negative root greater than τ h . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onepositive root smaller than − d/c and one negative root smallerthan or equal to τ h and greater than τ H . (iv) If cτ H + H ≤ − d , then there is one positive root x < − d/c and one negative root x ≤ τ H . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As c < γ < γ , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ > − δ >
0. Also: − δ < − δ < < cτ h + h < − δ
0, then there is one negative root x suchthat − λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d < − δ (pictured), then there is one negative root x such that ξ (3)2 < x ≤ λ and one non-negative root x ≤ min {− d/c, ξ (3)1 } . (iv) If − δ ≤ − d < − δ , then there is a negative root x such that ξ (2)2 ≤ x ≤ µ , another negative root x such that µ ≤ x < µ ,a third negative root x such that µ < x ≤ ξ (3)2 and one positiveroot x such that ξ (3)1 ≤ x < min { ξ (2)1 , − d/c } . (v) If − δ ≤ − d , then there is a positive root x such that ξ (2)1 ≤ x < − d/c and a negative root x ≤ ξ (2)2 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c and greater than τ h . (ii) If 0 ≤ − d < cτ h + h (pictured), then there is one non-negativeroot x ≤ − d/c , one negative root greater than τ h and either zeroor two negative roots smaller than − ( d + H ) /c (the latter appearwhen − δ ≤ cτ h + h ). (iii) If cτ h + h ≤ − d < cτ H + H , then there is one positive rootsmaller than − d/c , a negative root smaller than or equal to τ h and greater than τ H , and either zero or two negative roots smallerthan − ( d + H ) /c (the latter are always present if − δ ≤ cτ h + h ). (iv) If cτ H + H ≤ − d , then there is a positive root x < − d/c ,a negative root x < τ H and either zero or two negative rootssmaller than τ h and greater than or equal to − ( d + H ) /c . igure 4.11 a < b ≤ a , a > , c > , < c < γ < c < c < γ < c Notes As c < c < c , the quartic has three stationary points µ i and the number of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i .As γ < c < γ , the straight line − cx intersects x + ax + bx at three points: λ , , . Thus − δ < − δ >
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at two points: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c = c , one has − δ = − δ and cτ h + h = cτ H + H .As 0 < c < c , obviously − δ < − δ < < cτ h + h < cτ H + H < − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and also whether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one negative root x such that ξ (3)2 < x ≤ µ and another negative root x such that µ ≤ x < min { ξ (3)1 , − d/c } . (iii) If − δ ≤ − d <
0, then there are four negative roots: x such that λ < x ≤ µ , x such that µ ≤ x < λ , x such that λ < x ≤ ξ (3)2 , and x such that ξ (3)1 ≤ x < − d/c . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , another negative roor x such that λ ≤ x < µ , athird negative roor x such that µ < x ≤ λ and a positive root x ≤ min { ξ (2)1 , − d/c } . (v) If − δ ≤ − d (pictured), then there is a positive root x such that ξ (2)1 ≤ x < − d/c and a negative root x ≤ ξ (2)2 . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots, or there are two negative roots smaller than − d/c and greater than τ h , or there arefour negative roots, two of which smaller than − d/c and greater than τ h and the other two — smaller than − ( d + H ) /c . (ii) If 0 ≤ − d < cτ h + h , then there is one non-negative root x ≤ − d/c , and three negative roots: x > τ h and x , < − ( d + H ) /c . (iii) If cτ h + h ≤ − d < cτ H + H , then there is one positive root smaller than − d/c , a negative root smaller than or equal to τ h andgreater than τ H , and two negative roots smaller than − ( d + H ) /c . (iv) If cτ H + H ≤ − d (pictured), then there is a positive root x < − d/c , a negative root x < τ H and either zero or two negativeroots smaller than τ h and greater than or equal to − ( d + H ) /c . igure 4.12(continues on next page) a < b ≤ a , a > , c > , < c < γ < c < c < γ < c Notes (apply to all panes) As c < c < c , the quartic has three stationary points µ i and thenumber of real roots can be either 0, or 2, or 4. There are threetangents − cx − δ i to x + ax + bx — each at µ i (only shown onthe top-left pane).As γ < c < γ , the straight line − cx intersects x + ax + bx atthree points: λ , , . Thus − δ > − δ <
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c , one has − δ < − δ and also cτ H + H < cτ h + h .Top-left pane: one can have − δ < cτ H + H < − δ < < cτ h + h < − δ (pictured) or − δ < − δ ≤ cτ H + H < < cτ h + h < − δ .Top-right pane ( c closer to γ ): one can have − δ < cτ H + H
0, then there are four negative roots: x such that λ < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x < λ , x such that λ < x ≤ µ , and x such that µ ≤ x < − d/c . (iv) If 0 ≤ − d < − δ , then there is a negative root x such that ξ (2)2 < x ≤ λ , another negative root x such that λ ≤ x < µ , athird negative root x such that µ < x ≤ λ and a non-negative root x ≤ min { ξ (2)1 , − d/c } . (v) If − δ ≤ − d (pictured), then there is a positive root x such that ξ (2)1 ≤ x < − d/c and a negative root x ≤ ξ (2)2 . Analysis based on solving quadratic equations only (continues on next page)
Top-left pane(i) If − d < cτ H + H , then there are either no real roots, or there are two negative roots smaller than − ( d + H ) /c , or there are twonegative roots smaller than − ( d + H ) /c together with two negative roots greater than τ h and smaller than − d/c (the latter appearwhen cτ H + H ≥ − δ ). (ii) If cτ H + H ≤ − d <
0, then there is one negative root x such that − ( d + H ) /c ≤ x < − ( d + h ) /c , another negative root x < τ H and either zero or two negative roots smaller than τ h and greater than − d/c (the latter are always present if cτ H + H ≥ − δ , whilefor the pictured cτ H + H < − δ they may or may not be there). (iii) If 0 ≤ − d < cτ h + h , then there is one non-negative root x ≤ − d/c , a negative root x > τ h , another negative root x such that − ( d + H ) /c < x < − ( d + h ) /c , and a third negative root x < τ H . (iv) If cτ h + h ≤ − d (pictured), then there is one positive root x < − d/c , a negative root x < τ H and either zero or two negativeroots smaller than τ h and greater than or equal to − ( d + H ) /c . igure 4.12(continued from previous page) a < b ≤ a , a > , c > , < c < γ < c < c < γ < c Analysis based on solving quadratic equations only — continued from previous page
Top-right pane(i) If − d < cτ H + H , then there are either no real roots, or there are two negative roots smaller than − ( d + H ) /c , or there are twonegative roots smaller than − ( d + H ) /c together with two negative roots greater than τ h and smaller than − d/c (the latter appearwhen cτ H + H ≥ − δ ). (ii) If cτ H + H ≤ − d < cτ h + h , then there is one negative root x such that − ( d + H ) /c ≤ x < − ( d + h ) /c and another negativeroot x < τ H . If − δ > cτ h + h > cτ H + H , there are no other roots. If cτ h + h > cτ H + H ≥ − δ , there are two more positive rootsgreater than − d/c and smaller than τ h . If cτ h + h ≥ − δ > cτ H + H , these two roots may or may not be there. (iii) If cτ h + h ≤ − d <
0, then, if − δ ≤ cσ h + h , there will be a negative root x < τ H , two negative roots x , such that − ( d + H ) /c < x , ≤ τ h , and a negative root x such that σ h < x < − d/c . If however, cσ h + h < − δ , then the roots x and x may or may not be there. In summary: there is one negative root smaller than τ H and either one or three negative roots smaller than − d/c and greater than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is a non-negative root smaller than or equal to − d/c , either zero or two negative roots smallerthan τ h and greater than − ( d + H ) /c , and a negative root smaller than τ H . Bottom-right pane(i) If − d <
0, then there are either no real roots, or there are two negative roots smaller than τ H , or there are four negative roots:two smaller than τ H and two smaller than − d/c and greater than τ h . (ii) If 0 ≤ − d < cτ H + H , then there is one non-negative root smaller than − d/c , a neative root greater than τ h , and two negativeroots smaller than − ( d + H ) /c . (iii) If cτ H + H ≤ − d < cτ h + h , then there is one positive root x < − d/c , a negative root x > τ h , a negative root x such that − ( d + H ) /c ≤ x < − ( d + h ) /c and another negative root x < τ H . (iv) If cτ h + h ≤ − d (pictured), then there is a positive root smaller than − d/c , a negative root smaller than τ H and either zero ortwo negative roots smaller than τ h and greater than or equal to − ( d + H ) /c . igure 4.13 Figure 4.14 a < b ≤ a , a > , c > , a < b ≤ a , a > , c > , < c < γ < c < γ < c < c < c < γ < c < γ < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ). Thus − δ < − δ <
0. Also: − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c >
0, one has − δ < cσ H + H < − δ < cσ h + h < − δ < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one negative root x suchthat µ ≤ x < min { ξ (1)1 , − d/c } and another negative root x such that ξ (1)2 < x ≤ µ . (iii) If − δ ≤ − d < − δ , then there are four negative roots: x such that µ ≤ x < min { ξ (2)1 , − d/c } , x such that µ < x ≤ µ , x such that ξ (1)1 ≤ x < µ , and x such that ξ (2)2 < x ≤ ξ (1)2 . (iv) If − δ ≤ − d <
0, then there is a negative root x suchthat ξ (2)1 ≤ x < − d/c and another negative root x such that λ < x ≤ ξ (2)2 . (v) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots, or thereare two negative roots smaller than − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h , then there is a negative rootsmaller than τ H , a negative root greater than or equal to − ( d + H ) /c and smaller than − ( d + h ) /c , together with either zero ortwo negative roots greater than τ h and smaller than − d/c . (iii) If cτ h + h ≤ − d <
0, then there is a negative root greaterthan or equal to − ( d + h ) /c and smaller than − d/c , a negativeroot smaller than τ H , and either zero or two negative roots smallerthan or equal to τ h and greater than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is one non-negtive root x ≤ − d/c and one negative root x < τ H . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .As γ < γ < c , the straight line − cx intersects x + ax + bx atone point only ( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c >
0, one has − δ < cσ H + H < cσ h + h < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x suchthat µ ≤ x < − d/c and another negative root x such that λ < x ≤ µ . (iii) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots or thereare two negative roots smaller than − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h , then there is a negative roots x such that − ( d + H ) /c ≤ x < − ( d + h ) /c and another negativeroot x < τ H . (iii) If cτ h + h ≤ − d <
0, then there are two negative roots: x such that − ( d + h ) /c ≤ x < − d/c and x < τ H . (iv) If 0 ≤ − d (pictured), then there is one non-negtive root x ≤ − d/c and one negative root x < τ H . igure 5.1 Figure 5.2 a < b ≤ a , a < , c < , a < b ≤ a , a < , c < ,c < c < c < c < c < c < c < c < Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c <
0, one has − δ < cσ H + H < cσ h + h < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots or thereare two positive roots x , > − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h , then there are two positive roots: x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and x > τ H . (iii) If cτ h + h ≤ − d <
0, then there are two positive roots: x such that − d/c < x ≤ − ( d + h ) /c and x > τ H . (iv) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > τ H . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .The straight line − cx intersects x + ax + bx at one point only( λ ). Thus − δ < − δ <
0, and − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c <
0, one has − δ < − δ and also cσ H + H < cσ h + h .One can have either the pictured − δ < cσ H + H < − δ
0, then there is a positive root x suchthat − d/c < x ≤ ξ (2)2 and another positive root x such that ξ (2)1 ≤ x < λ . (v) If 0 ≤ − d (pictured), then there is a non-positive root x ≥− d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots, or thereare two positive roots x , > − ( d + H ) /c , or there are two positiveroots greater than − ( d + H ) /c together with two positive rootsgreater than − d/c and smaller than τ h (the latter appear when cτ H + H > − δ ). (ii) If cτ H + H ≤ − d < cτ h + h , then there are two positiveroots: x such that − ( d + h ) /c < x ≤ − ( d + H ) /c and x > τ H ,together with either zero or two positive roots x , such that − d/c < x , < τ h ( x , are always present if cτ H + H ≥ − δ ). (iii) If cτ h + h ≤ − d <
0, then there are two positive roots: x such that − d/c < x < − ( d + h ) /c and x > τ H , together witheither zero or two positive roots greater than or equal to τ h andsmaller than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > τ H . igure 5.4(Figure 5.3 is on next page) a < b ≤ a , a < , c < ,c < c < c < c < Notes (apply to all panes) As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c < c <
0, one can have − δ < cτ h + h < cτ H + H < c is closer to c ), or − δ < cτ h + h <
0, then there is one positive root x such that − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x ≥ λ . Analysis based on solving quadratic equations only (clockwise from the top-left pane, with four possibile situations per pane) (i) If − d < cτ h + h , then there are either no real roots or there are two positive roots greater than − d/c and smaller than τ h . (ii) If cτ h + h ≤ − d < cτ H + H , then there is a positive root x such that − d/c < x < − ( d + h ) /c and another positive root greaterthan or equal to τ h and smaller than τ H . (iii) If cτ H + H ≤ − d <
0, then there is a positive root greater than − d/c and smaller than − ( d + h ) /c and another positive rootgreater than or equal to τ H . (iv) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > τ H . (v) If − d < cτ h + h , then there are either no real roots or there are two positive roots greater than − d/c and smaller than τ h . (vi) If cτ h + h ≤ − d <
0, then there is a positive root x such that − d/c < x < − ( d + h ) /c and another positive root greater thanor equal to τ h and smaller than τ H . (vii) If 0 ≤ − d < cτ H + H , then there is a non-positive root x ≥ − d/c and a positive root x such that τ h < x < τ H . (viii) If cτ H + H ≤ − d (pictured), then there is one negative root x > − d/c and one positive root x ≥ τ H . (ix) If − d <
0, then there are either no real roots or there are two positive roots greater than − d/c and smaller than τ h . (x) If 0 ≤ − d < cτ h + h , then there is one non-positive root x ≥ − d/c and one positive root smaller than τ h . (xi) If cτ h + h ≤ − d < cτ H + H (pictured), then there is one negative root greater than − d/c and one positive root greater than orequal to τ h and smaller than τ H . (xii) If cτ H + H ≤ − d , then there is one negative root x > − d/c and one positive root x ≥ τ H . igure 5.3 Figure 5.5 a < b ≤ a , a < , c < , a < b ≤ a , a < , c > ,c < c < c < c < c < c < c < < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .The straight line − cx intersects x + ax + bx at one point only( λ ). Thus − δ < − δ <
0, and − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / > − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c < c <
0, one has − δ < − δ and also cσ h + h < cσ H + H .One can have either the pictured − δ < cσ h + h < − δ < cσ H + H < − δ < c is closer to c where µ and µ coalesce)or − δ < − δ < cσ h + h < cσ H + H < − δ < c is closerto c where cτ h + h and cτ h + H swap around).Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one positive root x suchthat max { ξ (1)2 , − d/c } < x ≤ µ and another positive root x such that µ ≤ x < ξ (1)1 . (iii) If − δ ≤ − d < − δ , then there are four positive roots: x such that max { ξ (2)2 , − d/c } < x ≤ ξ (1)2 , x such that ξ (1)1 ≤ x <µ , x such that µ < x ≤ µ , and x such that µ ≤ x < ξ (2)1 . (iv) If − δ ≤ − d <
0, then there is a positive root x suchthat − d/c < x ≤ ξ (2)2 and another positive root x such that ξ (2)1 ≤ x < λ . (v) If 0 ≤ − d (pictured), then there is a non-positive root x ≥− d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d < cτ h + h , then there are either no real roots, or thereare two positive roots greater than − d/c and smaller than τ h , orthere are two positive roots greater than − d/c and smaller than τ h , together with two positive roots greater than − ( d + H ) /c (thelatter appear when cτ h + h > − δ ). (ii) If cτ h + h ≤ − d < cτ H + H , then there is a positive root x such that − d/c < x < − ( d + h ) /c , a positive root x such that τ h ≤ x < τ H , and either zero or two positive roots greater than − ( d + H ) /c (the latter are always present of cτ h + h ≥ − δ ). (iii) If cτ H + H ≤ − d <
0, then there is a positive root x suchthat − d/c < x < − ( d + h ) /c , a positive root x > τ H , and eitherzero or two positive roots smaller than or equal to − ( d + H ) /c and greater than τ h . (iv) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > τ H . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x such that λ < x ≤ µ and another negative root x such that µ ≤ x < − d/c . (iii) If 0 ≤ − d (pictured), then there is a negative root x ≤ λ and a non-negative root x ≤ − d/c . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots smaller than − d/c . (ii) If 0 ≤ − d < cτ h + h , then there is a non-positive root x and a non-negative root x such that max { , − ( d + h ) /c } ≤ x < min {− d/c, τ h } . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onenegative root x and one positive root x such that max { τ h , − ( d + H ) /c } ≤ x < min {− ( d + h ) /c, τ H } . (iv) If cτ H + H ≤ − d , then there is a negative root x and apositive root x such that τ H ≤ x < − ( d + H ) /c . igure 5.7(Figure 5.6 is on next page) a < b ≤ a , a > , c > , < c < c < c < c Notes (apply to all panes) As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > < c < c < c , one can have − δ < < cτ h + h < cτ H + H (top-left pane, when c is closer to 0), or − δ < cτ h + h <
0, then there is one negative root x such that µ ≤ x < − d/c and another negative root x such that λ < x ≤ µ . (iii) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x ≤ λ . Analysis based on solving quadratic equations only (clockwise from the top-left pane, with four possibile situations per pane) (i) If − d <
0, then there are either no real roots or there are two negative roots smaller than − d/c and greater than τ h . (ii) If 0 ≤ − d < cτ h + h , then there is one non-negative root x ≤ − d/c and one negative root greater than τ h . (iii) If cτ h + h ≤ − d < cτ H + H , then there is one positive root smaller than − d/c and one negative root smaller than or equal to τ h and greater than τ H . (iv) If cτ H + H ≤ − d (pictured), then there is one positive root x < − d/c and one negative root x ≤ τ H . (v) If − d < cτ h + h , then there are either no real roots or there are two negative roots smaller than − d/c and greater than τ h . (vi) If cτ h + h ≤ − d <
0, then there is a negative root x such that − ( d + h ) /c < x < − d/c and another negative root smaller thanor equal to τ h and greater than τ H . (vii) If 0 ≤ − d < cτ H + H , then there is a non-negative root x ≤ − d/c and a negative root x such that τ H < x < τ h . (viii) If cτ H + H ≤ − d (pictured), then there is one positive root x < − d/c and one negative root x ≤ τ H . (ix) If − d < cτ h + h , then there are either no real roots or there are two negative roots smaller than − d/c and greater than τ h . (x) If cτ h + h ≤ − d < cτ H + H , then there is a negative root x such that − ( d + h ) /c < x < − d/c and another negative root smallerthan or equal to τ h and greater than τ H . (xi) If cτ H + H ≤ − d <
0, then there is a negative root smaller than − d/c and greater than − ( d + h ) /c and another negative rootsmaller than or equal to τ H . (xii) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x < τ H . igure 5.6 Figure 5.8 a < b ≤ a , a > , c < , a < b ≤ a , a > , c > ,c < < c < c < c < c < c < c < c Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).Obviously, cτ H + H > cτ h + h > > − δ .Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots greater than − d/c . (ii) If 0 ≤ − d < cτ h + h , then there is a non-negative root x and a non-positive root x such that max {− d/c, τ h } < x ≤ min { , − ( d + h ) /c } . (iii) If cτ h + h ≤ − d < cτ H + H (pictured), then there is onepositive root x and one negative root x such that max {− ( d + h ) /c, τ H } < x ≤ min { τ h , − ( d + H ) /c } . (iv) If cτ H + H ≤ − d , then there is a positive root x and anegative root x such that − ( d + H ) /c < x ≤ τ H . Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .The straight line − cx intersects x + ax + bx at one point only( λ ). Thus − δ < − δ <
0, and − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c >
0, one has − δ < − δ and also cσ h + h < cσ H + H .One can have either the pictured − δ < cσ h + h < − δ < cσ H + H < − δ < c is closer to c where µ and µ coalesce)or − δ < − δ < cσ h + h < cσ H + H < − δ < c is closerto c where cτ h + h and cτ h + H swap around).Consideration of whether − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d < − δ , then there is one negative root x suchthat µ ≤ x < min { ξ (3)1 , − d/c } and another negative root x such that ξ (3)2 < x ≤ µ . (iii) If − δ ≤ − d < − δ , then there is a negative root x such that ξ (3)1 ≤ x < min {− d/c, ξ (2)1 } , another negative root x such that µ < x ≤ ξ (3)2 , a third negative root x such that µ ≤ x < µ ,and a fourth negative root x such that ξ (2)2 < x ≤ µ . (iv) If − δ ≤ − d <
0, then there is a negative root x suchthat ξ (2)1 ≤ x < − d/c and another negative root x such that λ < x ≤ ξ (2)2 . (v) If 0 ≤ − d (pictured), then there is a non-negative root x ≤− d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cτ h + h , then there are either no real roots, or thereare two negative roots smaller than − d/c and greater than τ h , orthere are two negative roots smaller than − d/c and greater than τ h , together with two negative roots smaller than − ( d + H ) /c (thelatter appear when cτ h + h > − δ ). (ii) If cτ h + h ≤ − d < cτ H + H , then there is a negative root x such that − ( d + h ) /c < x < − d/c , a negative root x such that τ H < x ≤ τ h , and either zero or two negative roots smaller than − ( d + H ) /c (the latter are always present of cτ h + h ≥ − δ ). (iii) If cτ H + H ≤ − d <
0, then there is a negative root x suchthat − ( d + h ) /c < x < − d/c , a negative root x < τ H , and eitherzero or two negative roots greater than or equal to − ( d + H ) /c and smaller than τ h . (iv) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x > τ H . igure 5.9 Figure 5.10 a < b ≤ a , a > , c > , a < b ≤ a , a > , c > , < c < c < c < c < c < c < c < c Notes As c < c < c , the quartic has three stationary points µ i andthe number of real roots can be either 0, or 2, or 4. There arethree tangents − cx − δ i to x + ax + bx — each at µ i .The straight line − cx intersects x + ax + bx at one point only( λ ). Thus − δ < − δ <
0, and − δ < τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c >
0, one has − δ < − δ and also cσ H + H < cσ h + h .One can have either the pictured − δ < cσ H + H < − δ
0, then there is a negative root x suchthat ξ (2)1 ≤ x < − d/c and another negative root x such that λ < x ≤ ξ (2)2 . (v) If 0 ≤ − d (pictured), then there is a non-negative root x ≤− d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots, or thereare two negative roots smaller than − ( d + H ) /c , or there are twonegative roots smaller than − ( d + H ) /c together with two negativeroots greater than τ h and smaller than − d/c (the latter appearwhen cτ H + H > − δ ). (ii) If cτ H + H ≤ − d < cτ h + h , then there are two negative roots: x such that − ( d + H ) /c ≤ x < − ( d + h ) /c , and x < τ H witheither zero or two negative roots x , such that τ h < x , < − d/c ( x , are always present if cτ H + H ≥ − δ ). (iii) If cτ h + h ≤ − d <
0, then there is a negative root x suchthat − ( d + h ) /c ≤ x < − d/c , another negative root x < τ H ,and either zero or two negative roots smaller than or equal to τ h and greater than − ( d + H ) /c . (iv) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x < τ H . Notes As c < c < c , the quartic has a single stationary point µ andthe number of real roots can be either 0 or 2. There is only onetangent − cx − δ to x + ax + bx — the one at µ .The straight line − cx intersects x + ax + bx at one point only( λ ).The straight line joining the two points of curvature change ( τ h and τ H ) has the same slope − c = − (1 / a ( b − a / < − c x − d , tangent to x + ax + bx at twopoints: α and β = − α − a/
2, where α and β are the simultaneousdouble roots of the varied quartic x + ax + bx + c x + d [with d = (1 / b − a / > c > c >
0, one has − δ < cσ H + H < cσ h + h < − d ≤ cτ h + h or − d > cτ h + h and alsowhether − d ≤ cτ H + H or − d > cτ H + H would yield sharperbounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x suchthat µ ≤ x < − d/c and another negative root x such that λ < x ≤ µ . (iii) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only (i) If − d < cτ H + H , then there are either no real roots, or thereare two negative roots x , < − ( d + H ) /c . (ii) If cτ H + H ≤ − d < cτ h + h , then there are two negative roots: x such that − ( d + H ) /c ≤ x < − ( d + h ) /c and x < τ H . (iii) If cτ h + h ≤ − d <
0, then there are two negative roots: x such that − ( d + h ) /c ≤ x < − d/c and x < τ H . (iv) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x < τ H . igure 6.1 b > a , a < , c < Notes (apply to all panes)
The quartic has a single stationary point µ and the number ofreal roots can be either 0 or 2. There is only one tangent − cx − δ to x + ax + bx — the one at µ . The intersection point of thistangent with the ordinate is − δ and this is always negative.The straight line − cx intersects x + ax + bx only at λ > µ .The third derivative of x + ax + bx vanishes at φ = − a/ x + ax + bx .The “separator” straight line with equation − c ( x + a/
4) +( a / b − (3 / a /
4] passes through the “marker” point φ andintersects the ordinate at point t which is always greater than orequal to − δ . This point t is zero if c = ζ ≡ ( a/ b − (3 / a /
4] = a /
64 + c /
2, where c = (1 / a ( b − a / ζ <
0, when a <
0. Also: ζ − c = − ( a/ b − (5 / a /
4] and thus ζ > c for a < b > (3 / a / µ = φ if c = c = (1 / a ( b − a / a < b > (3 / a /
4, then c <
0. Hence, at c = c , one has t = − δ = − ( a / b − (5 / a / <
0, since b > (3 / a / c < c <
0, one has µ > φ and thus − δ < t < c < c <
0, one has µ < φ . There are two possibilities in thiscase: either the pictured on the top-right pane − δ < t ≤
0, which occurs for c < c ≤ ζ <
0, or the pictured on the bottom-rightpane − δ < < t , which occurs for c < ζ < c < − d ≤ t or − d > t would yield sharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (applies to all panes) (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x such that − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only
Top-left pane: c < c < ζ <
0, hence µ > φ and − δ < t < (i) If − d < t , then there are either no real roots, or there are two positive roots x , > − ( d + T ) /c . (ii) If t ≤ − d <
0, then there are two positive roots: x such that − d/c < x ≤ − ( d + T ) /c and x > φ = − a/ (iii) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > φ = − a/ c < c ≤ ζ <
0, hence µ < φ and − δ < t ≤ (i) If − d < t , then there are either no real roots, or there are two positive roots x , such that − d/c < x , < φ = − a/ (ii) If t ≤ − d <
0, then there is a positive root x such that − d/c < x ≤ − ( d + T ) /c and a positive root x ≥ φ = − a/ (iii) If 0 ≤ − d (pictured), then there is one non-positive root x ≥ − d/c and one positive root x > φ = − a/ c < ζ < c <
0, hence µ < φ and − δ < < t (i) If − d <
0, then there are either no real roots, or there are two positive roots x , such that − d/c < x , < φ = − a/ (ii) If 0 ≤ − d < t , then there is one non-positive root x ≥ − d/c and a positive root x < φ = − a/ (iii) If t ≤ − d (pictured), then there is one negative root x > − d/c and one positive root x ≥ φ = − a/ igure 6.2 Figure 6.3 b > a , a < , c > b > a , a > , c < Notes
The quartic has a single stationary point µ and the number ofreal roots can be either 0 or 2. There is only one tangent − cx − δ to x + ax + bx — the one at µ . The intersection point of thistangent with the ordinate is − δ and this is always negative.The straight line − cx intersects x + ax + bx only at λ < µ .The third derivative of x + ax + bx vanishes at φ = − a/ x + ax + bx .The “separator” straight line with equation − c ( x + a/
4) +( a / b − (3 / a /
4] passes through the “marker” point φ andintersects the ordinate at point t > − cx +( a / b − (3 / a /
4] provides sharper bounds for the analysis.It intersects the ordinate at point T = t + ac/ < t , since a < c > − δ < < T < t .Consideration of whether − d ≤ t or − d > t and also whether − d ≤ T or − d > T would yield sharper bounds in the analysisbased on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x suchthat µ ≤ x < − d/c and another negative root x such that λ < x ≤ µ . (iii) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x ≤ λ . Notes
The quartic has a single stationary point µ and the number ofreal roots can be either 0 or 2. There is only one tangent − cx − δ to x + ax + bx — the one at µ . The intersection point of thistangent with the ordinate is − δ and this is always negative.The straight line − cx intersects x + ax + bx only at λ > µ .The third derivative of x + ax + bx vanishes at φ = − a/ x + ax + bx .The “separator” straight line with equation − c ( x + a/
4) +( a / b − (3 / a /
4] passes through the “marker” point φ andintersects the ordinate at point t > − cx +( a / b − (3 / a /
4] provides sharper bounds for the analysis.It intersects the ordinate at point T = t + ac/ < t , since a > c < − δ < < T < t .Consideration of whether − d ≤ t or − d > t and also whether − d ≤ T or − d > T would yield sharper bounds in the analysisbased on solving cubic equations. Analysis based on solving cubic equations (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one positive root x suchthat − d/c < x ≤ µ and another positive root x such that µ ≤ x < λ . (iii) If 0 ≤ − d (pictured), then there is a non-positive root x ≥ − d/c and a positive root x ≥ λ . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twonegative roots x , < − d/c . (ii) If 0 ≤ − d < T (pictured), then there is one non-negative root x ≤ − d/c and one negative root x . (iii) If T ≤ − d < t , then there is one positive root x such that − ( d + T ) /c < x < φ = − a/ x . (iv) If t ≤ − d , then there is one positive root x such that φ = − a/ ≤ x < − ( d + T ) /c and one negative root x . Analysis based on solving quadratic equations only (i) If − d <
0, then there are either no real roots or there are twopositive roots x , > − d/c . (ii) If 0 ≤ − d < T , then there is one non-positive root x ≥ − d/c and one positive root x . (iii) If T ≤ − d < t (pictured), then there is one negative root x such that φ = − a/ < x < − ( d + T ) /c and one positive root x . (iv) If t ≤ − d , then there is one negative root x such that − ( d + T ) /c < x ≤ φ = − a/ x . igure 6.4 b > a , a > , c > Notes (apply to all panes)
The quartic has a single stationary point µ and the number ofreal roots can be either 0 or 2. There is only one tangent − cx − δ to x + ax + bx — the one at µ . The intersection point of thistangent with the ordinate is − δ and this is always negative.The straight line − cx intersects x + ax + bx only at λ > µ .The third derivative of x + ax + bx vanishes at φ = − a/ x + ax + bx .The “separator” straight line with equation − c ( x + a/
4) +( a / b − (3 / a /
4] passes through the “marker” point φ andintersects the ordinate at point t which is always greater than orequal to − δ . This point t is zero if c = ζ ≡ ( a/ b − (3 / a /
4] = a /
64 + c /
2, where c = (1 / a ( b − a / ζ >
0, when a >
0. Also: ζ − c = − ( a/ b − (5 / a /
4] and thus ζ < c for a > b > (3 / a / µ = φ if c = c = (1 / a ( b − a / a > b > (3 / a /
4, then c >
0. Hence, at c = c , one has t = − δ = − ( a / b − (5 / a / <
0, since b > (3 / a / c > c >
0, one has µ < φ and thus − δ < t < c > c >
0, one has µ > φ . There are two possibilities in thiscase: either the pictured on the top-right pane − δ < t ≤
0, which occurs for c > c ≥ ζ >
0, or the pictured on the bottom-rightpane − δ < < t , which occurs for c > ζ > c > − d ≤ t or − d > t would yield sharper bounds in the analysis based on solving cubic equations. Analysis based on solving cubic equations (applies to all panes) (i) If − d < − δ , then there are no real roots. (ii) If − δ ≤ − d <
0, then there is one negative root x such that µ ≤ x < − d/c and another negative root x such that λ < x ≤ µ . (iii) If 0 ≤ − d (pictured), then there is a non-negative root x ≤ − d/c and a negative root x ≤ λ . Analysis based on solving quadratic equations only
Top-left pane: c > c > ζ >
0, hence µ < φ and − δ < t < (i) If − d < t , then there are either no real roots, or there are two negative roots x , < − ( d + T ) /c . (ii) If t ≤ − d <
0, then there are two negative roots: x such that − ( d + T ) /c ≤ x < − d/c and x < φ = − a/ (iii) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x < φ = − a/ c > c ≥ ζ >
0, hence µ > φ and − δ < t ≤ (i) If − d < t , then there are either no real roots, or there are two negative roots x , such that φ = − a/ < x , < − d/c . (ii) If t ≤ − d <
0, then there is a negative root x such that − ( d + T ) /c ≤ x < − d/c and a negative root x ≤ φ = − a/ (iii) If 0 ≤ − d (pictured), then there is one non-negative root x ≤ − d/c and one negative root x < φ = − a/ c > ζ > c >
0, hence µ > φ and − δ < < t (i) If − d <
0, then there are either no real roots, or there are two negative roots x , such that φ = − a/ < x , < − d/c . (ii) If 0 ≤ − d < t , then there is one non-negative root x ≤ − d/c and a negative root x > φ = − a/ (iii) If t ≤ − d (pictured), then there is one positive root x < − d/c and one negative root x ≤ φ = − a/ cknowledgements It is a pleasure to thank Elena Tonkova for useful discussions and Milena E. Mihaylovafor the help with the figures.
References [1] Emil M. Prodanov,