# A proof about maximum point, critical point and differentiation

• March 4th 2013, 07:57 PM
ianchenmu
A proof about maximum point, critical point and differentiation
Let $E\subset$ $\mathbb{R}^n$ and $f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $a$is a local maximum point for $f$, then either $f$ is differentiable at $x = a$ and $Df(a) = 0$ or $f$ is not differentiable at $x = a$. Deduce that if $f$ is differentiable on $E^o$, then a global maximum point of f is either a critical point of f or an element of $\partial E$.

It's a little bit about optimization but stil analysis. Well I have no idea about this question and I think I need a proof. Thank you!