Second Derivative Test for Local Extrema

Test

Suppose that $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ is a class of $C^2$ Class Functions Suppose $x_0$ is a Critical Point

• $Hf(x_0)$ is Positive Definite Matrix $⟹$ $x_0$ is a relative minimum
• $Hf(x_0)$ is Negative Definite Matrix $⟹$ $x_0$ is a relative maximum
• $Hf(x_0)$ is neither $⟹$ no conclusion, Saddle Point