Let $f\colon (a,b) \to \mathbb{R}$ a non constant differentiable function.
Is the following statement true:
If $f$ has a local maximum and a local minimum then $f$ also does have an inflection point.
If so, how to prove it, if not, what would be a counterexample?
Remark
If there are $a_0,b_0 \in [a,b]$ with $a_0 < x_0 < b_0$ such that $f|_{(a_0,x_0)}$ is convex and $f|_{(x_0,b_0)}$ concave or vice versa, then $(x_0,f(x_0))$ is called inflection point of $f$ (Amann Escher Analysis I, p349).
Sometimes stricly concave (convex) is used in the definition. Does this change the theorem?
If the theorem is not true, does it hold if one allows only smooth functions or even more restrictive only polynomial ones (excluding linear functions)?
Best Answer
Consider this:
$$ f(x) = \begin{cases} (x+1)(x+3)-2 & \text{when }x\le -1 \\ 2x & \text{when } -1 < x < 1 \\ 2-(x-1)(x-3) & \text{when } x \ge 1 \end{cases} $$
(Graphed by Wolfram Alpha)
This has local minimum and maximum at $x=-2$ and $x=2$ respectively, but would you consider it to have an inflection point? Different defintions would give different answers.
In the definition you quote, every point in $[-1,1]$ would be an inflection point if you have "convex" and "concave"; but no points would be inflection point if you have "strictly convex/concave".
A counterexample to the claim in the non-strict case would be the indefinite integral of the Weierstrass function. It's easy to see that it must have local maxima and minima somewhere (e.g. you can find closed intervals where the endpoints are certainly not maxima, but a continuous function always attains its supremum over a closed interval). On the other hand the Weierstrass function itself is nowhere monotone, so its integral cannot be convex or concave on any interval, which again excludes the existence of inflection points.
Even requiring that $f$ is twice differentiable won't do; according to this anwswer there exist everywhere differentiable yet nowhere monotone functions, and (with an offset to make sure they cross the $x$ axis) their indefinite integrals would be counterexamples.