# Thread: Fermat's theorem (stationary points) of higher dimensions

1. ## Fermat's theorem (stationary points) of higher dimensions

Look at this page and the Proof part,

Fermat's theorem (stationary points) - Wikipedia, the free encyclopedia

How to change the proof 2 into a proof of higher dimensions or can you give a proof of Fermat's theorem of higher dimensions?

To clarify, I need to prove this:
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 $x=a$ with $Df(a)=0$ or $f$ is not differentiable at $a$.

2. ## Re: Fermat's theorem (stationary points) of higher dimensions

Df(x) is, for each x, a vector (or can be represented by such a vector, depending upon exactly how you are defining Df(x)) pointing in the direction of fastest increase. Show that if f is differentiable at a and Df(a) is not 0 then moving a short distance in the direction of Df(a) will give a higher value of f and in the opposite direction will give a lower value. You can use the mean value theorem for that.