# Even functions and inflection points

Can an even function have an inflection point? If yes, then give an example while if not then give a proof. If needed you may assume that $f$ is two times differentiable .

Intution says that even functions don't have inflection points, but I cannot settle down an appropriate proof.

On the contrary for an odd function this is possible. For instance $f(x)=x^3$ has an inflection point at $x=0$.

• Does it have to be at $x=0$? – mickep Sep 15 '15 at 18:22
• Thank you all for your answers. – Tolaso Sep 15 '15 at 18:58

$\cos\colon\mathbb{R}\to\mathbb{R}$ is even and it has infinitely many inflection points (at its zeros, which are found at $\{n\pi\}_{n\in\mathbb{Z}}$).
Consider $$f(x)=\max\bigl((x-1)^3,-(x+1)^3\bigr).$$ Or, for a nicer example, consider $$g(x)=(x-1)^3(x+1)^3.$$
the function $y=x^4-4x^2$ has at least one inflection point WA plot