TITLE: In polynomial functions, is the midpoint of two adjacent minimum and maximum always a point of inflection?
QUESTION [15 upvotes]: For any generic polynomial function, it seems that a point of inflection is always half-way between the nearest minimum and maximum (if they exist).
Is there a way to prove that midpoints of adjacent minima and maxima are POIs? Or might I be missing a simple counterexample?

REPLY [2 votes]: It's easy to prove that this cannot be the case, by substituting $x=u^2$. Clearly your polynomial is still a polynomial, although now with only even powers of $u$. This substitution is non-linear, so a point $x_{mid} = (x_a+x_b)/2$ will not correspond to $u_{mid} = (u_a+u_b)/2$. 
So if $x_{mid}$ is an inflection point, then $u_{mid}$ isn't, and vice versa. And that means there's at least one polynomial for which the assertion doesn't hold.