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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 $∂E$.

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!

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!

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