Let $f$ be differentiable at every point of some open ball $B(a)$ in $\mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then prove $D_k f(a)=0$.

Question

If $f:\mathbb R^n \to \mathbb R$ is a function differentiable at every point of some open ball $B(a)$ with center $a\in \mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then how to show that all partial derivatives of $f$ evaluated at $a$ is $0$ ?

Answer 1

Hints. Let $\newcommand{\e}{\mathbf{e}}\e_1,\ldots,\e_n$ be the cvectors of the standar basis of $\Bbb R^n$.

There's a $\delta \gt 0$ such that for each $t\in I:= ({-}\delta,\delta)$ and each $j\in\{1,\dots,n\}$, the point $a+t\e_j\in B(a)$.

Then for each $j$ consider the function $g_j:I\to \Bbb R$ given by $$g(t) = f(a+t\e_j).$$

Use what you know from analysis in one real variable.