09 - Gradient Vector on Critical Points

Gradient Vector on Critical Points

Suppose that $f(x,y)$ is defined on an open subset $U$ with

• continuous partial derivatives
• $(x_0,y_0)\in U$ being a local maximum or local minimum

Then

$$\frac{\partial f}{\partial x}(x_0,y_0)=\frac{\partial f}{\partial y}(x_0,y_0)=0$$

Hence the gradient vector $\nabla f(x_0,y_0)=0$

Proof