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?
Best Answer
You are probably only trying cubics. If you have the function $f(x)=ax^3+bx^2+cx+d$ the local extrema come at the roots of $3ax^2+2bx+c=0$, which are $\frac {-b\pm \sqrt{b^2-3ac}}{3a}$. The midpoint of these is $\frac {-b}{3a}$. The inflection point is at the root of $6ax+2b=0$, which is $\frac {-b}{3a}$
If you try higher degree polynomials it fails. Take $f(x)=x^4-2x^2+1$ The extrema are at roots of $4x^3-4x=0$, so at $0,\pm 1$. The inflection points are at the roots of $12x^2-4=0$, which are $\pm \frac 1{\sqrt 3}$, not $\pm \frac12$