First Derivative Test for Local Extrema

Test

Suppose $U\subset {\mathbb{R}}^n$ is an Open Set Suppose $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ is Differentiable If $x_0\in U$ is a Local Extrema, then $Df(x_0)=0$

Intuition

Local extrema and differentiable $⟹$ critical point