Gradient Perpendicular to Level Sets

Theorem

• Suppose that $f:{\mathbb{R}}^n\to \mathbb{R}$ is Differentiable at $x_0$
• Suppose $\nabla f(x_0)\ne 0$
• Let $c=f(x_0)$
• Consider Level Surface $L_c=\{x:c=f(x)\}\subset {\mathbb{R}}^n$
• Then, the Gradient $\nabla f(x_0)$ is perpendicular to the tangent plane $L_c$ at $x_0$