Polynomial function/Degree d/Maximal number of maxima and minima/Exercise

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be a polynomial function of degree ${}d\geq 1$. Let ${}m$ be the number of local maxima of ${}f$ and ${}n$ the number of local minima of ${}f$. Prove that if ${}d$ is odd then ${}m=n$ and that if ${}d$ is even then

${}\vert {m-n}\vert =1\,.$

Create a solution