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Say I'm given a general sextic equation $$ax^6+bx^5+cx^4+dx^3+ex^2+fx+g =0$$

I want to know how I can tell where the critical points (or least say how many there will be) will be since the Abel-Ruffini theorem tells me that it is not possible to express the solutions via radicals, because the derivative of the above functions will be a $5^{th}$ order polynomial. I'd also like to know if there is away to determine how the extrema will change if I vary the parameters. Anyone knows some methods that make this possible? I know about Descartes rule of signs but that doesn't give me a very clear answer just a range. I also know that the roots of an arbitrary sextic equation can be expressed as the soluution of a hypergeometric differential equation but I don't want something that deep.

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  • $\begingroup$ The best I can come up with: There are either $1,3$ or $5$ critical points using the conjugate root theorem (assuming that the polynomial have real coefficients. $\endgroup$ – Yiyuan Lee Apr 6 '14 at 5:30
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As you say, the derivative is a polynomial of degree $5$ with symbolic coefficients. The discriminant of that derivative is, according to Maple, $$4050000\,{a}^{4}{f}^{4}-5400000\,{a}^{3}be{f}^{3}-9720000\,{a}^{3}cd{f }^{3}+6912000\,{a}^{3}c{e}^{2}{f}^{2}+8748000\,{a}^{3}{d}^{2}e{f}^{2}- 8294400\,{a}^{3}d{e}^{3}f+1769472\,{a}^{3}{e}^{5}+5400000\,{a}^{2}{b}^ {2}d{f}^{3}-180000\,{a}^{2}{b}^{2}{e}^{2}{f}^{2}+6480000\,{a}^{2}b{c}^ {2}{f}^{3}-8856000\,{a}^{2}bcde{f}^{2}+921600\,{a}^{2}bc{e}^{3}f- 4374000\,{a}^{2}b{d}^{3}{f}^{2}+6609600\,{a}^{2}b{d}^{2}{e}^{2}f- 1658880\,{a}^{2}bd{e}^{4}-4147200\,{a}^{2}{c}^{3}e{f}^{2}+4276800\,{a} ^{2}{c}^{2}{d}^{2}{f}^{2}+3870720\,{a}^{2}{c}^{2}d{e}^{2}f-1179648\,{a }^{2}{c}^{2}{e}^{4}-4898880\,{a}^{2}c{d}^{3}ef+1492992\,{a}^{2}c{d}^{2 }{e}^{3}+944784\,{a}^{2}{d}^{5}f-314928\,{a}^{2}{d}^{4}{e}^{2}-4800000 \,a{b}^{3}c{f}^{3}+720000\,a{b}^{3}de{f}^{2}-216000\,a{b}^{3}{e}^{3}f+ 4896000\,a{b}^{2}{c}^{2}e{f}^{2}+3024000\,a{b}^{2}c{d}^{2}{f}^{2}- 5371200\,a{b}^{2}cd{e}^{2}f+1382400\,a{b}^{2}c{e}^{4}+194400\,a{b}^{2} {d}^{3}ef-64800\,a{b}^{2}{d}^{2}{e}^{3}-3628800\,ab{c}^{3}d{f}^{2}+ 184320\,ab{c}^{3}{e}^{2}f+3075840\,ab{c}^{2}{d}^{2}ef-921600\,ab{c}^{2 }d{e}^{3}-699840\,abc{d}^{4}f+233280\,abc{d}^{3}{e}^{2}+663552\,a{c}^{ 5}{f}^{2}-663552\,a{c}^{4}def+196608\,a{c}^{4}{e}^{3}+165888\,a{c}^{3} {d}^{3}f-55296\,a{c}^{3}{d}^{2}{e}^{2}+800000\,{b}^{5}{f}^{3}-960000\, {b}^{4}ce{f}^{2}-720000\,{b}^{4}{d}^{2}{f}^{2}+1080000\,{b}^{4}d{e}^{2 }f-270000\,{b}^{4}{e}^{4}+864000\,{b}^{3}{c}^{2}d{f}^{2}-48000\,{b}^{3 }{c}^{2}{e}^{2}f-720000\,{b}^{3}c{d}^{2}ef+216000\,{b}^{3}cd{e}^{3}+ 162000\,{b}^{3}{d}^{4}f-54000\,{b}^{3}{d}^{3}{e}^{2}-172800\,{b}^{2}{c }^{4}{f}^{2}+172800\,{b}^{2}{c}^{3}def-51200\,{b}^{2}{c}^{3}{e}^{3}- 43200\,{b}^{2}{c}^{2}{d}^{3}f+14400\,{b}^{2}{c}^{2}{d}^{2}{e}^{2} $$ As you vary the coefficients, the points where the number of real critical points changes will be points where that discriminant is $0$.

You can also use a Sturm sequence to determine the number of real critical points, or the number in a given interval, for given values of the coefficients.

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  • $\begingroup$ Wow... that's hideous lol. What command in MAPLE did you use to find the discriminant? $\endgroup$ – Millardo Peacecraft Apr 6 '14 at 5:26
  • $\begingroup$ discrim(diff(p,x),x); $\endgroup$ – Robert Israel Apr 6 '14 at 18:39

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