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 $x=0$.