Does Fermat’s stationary points theorem generalize to multivariable case?

Question

Let $G$ be an open subset of a real-normed space $X$, and $f:X\rightarrow \mathbb{R}$ be a Frechet differentiable function.

Assume that $f$ has a local extremum at a point $M\in G$. Then, is $Df(M)=0$?

This is true when $X$ is finite dimensional, but is this still true if $X$ is infinite dimensional?

score 1 · Accepted Answer · answered Dec 03 '17 at 18:51

Let $p$ be the extremum. Consider $\gamma_v:\mathbb{R}\to X$ a differentiable curve such that $\gamma_v(0) = p$ and $\gamma_v'(0)=v \in X$. Then $F\circ \gamma_v:\mathbb{R}\to X$ is Frechet differentiable and $dF\circ \gamma_v(0) = dF(p)\circ d\gamma_v(0) = dF(p)(v) $. Since $p$ is a local extremum also for $F\circ \gamma_v$ we have that $0= dF\circ \gamma_v(0)=dF(p)v$ for any $v\in X$.

Does Fermat’s stationary points theorem generalize to multivariable case?

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