Second-Derivative Test

Definition

Suppose that $f$ and its first and second derivatives are continuous throughout a disk centered at $P_0$ and $\nabla f(P_0)=0$.

• If the Hessian Matrix matrix is a positive-definite, then $P_0$ is a Local Minimum ${\nabla }^{2}f(P_0)\succ 0$
• If the Hessian Matrix matrix is a negative-definite, then $P_0$ is a Local Maximum ${\nabla }^{2}f(P_0)\prec 0$ If the Hessian Matrix matrix contains both positive and negative eigenvalues, then the $P_0$ is a Saddle Point