Keep it (mostly) real

Calculus Level 5

$f(x)=2x^6+6x^5-15x^4-40x^3+120x^2+ax+b$

How many values of $a$ are there such that $f(x)$ has at least four real roots, counted with their multiplicities?

3 1 0 Infinitely many 6 2

1 solution

Otto Bretscher
Mar 22, 2016

Note that $f''(x)=60(x-1)^2(x+2)^2\geq 0$ , so that $f(x)$ is convex . A convex function either has up to two simple real roots (not enough for us) or it has a single real root $c$ of even multiplicity $m$ . Since $f''(x)$ has the two double roots 1 and -2, we must have $m=4$ and $c=1$ or $c=-2$ . Thus there are $\boxed{2}$ such functions.

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Interesting problem based on the observation of convex functions.

Keep it (mostly) real

1 solution

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