Let $x$ be a vector, and let $H(x)$ be the Hessian for $f$ at a point $x$ . Let $f$ have continuous partial derivatives of first and second order in a neighborhood of $x$ . Let $\nabla f (x)= 0$ .

If $H(x)$ is positive definite, then $x$ is a strict local minimum for $f$ .

If $x$ is a local minimum for $x$ , then $H(x)$ is positive semidefinite.

If $H(x)$ is negative definite, then $x$ is a strict local maximum for $f$ .

If $x$ is a local maximum for $x$ , then $H(x)$ is negative semidefinite.

If $H(x)$ is indefinite, $x$ is a nondegenerate saddle point.

If the case when the dimension of $x$ is 1 (i.e. $f: \mathbb{R} \to \mathbb{R}$ ), this reduces to the Second Derivative Test, which is as follows:

Let the neighborhood of $x$ be in the domain for $f$ , and let $f$ have continuous partial derivatives of first and second order. Let $f'(x) = 0$ . If $f''(x) > 0$ , then $x$ is a strict local minimum. If $f''(x) < 0$ , then $x$ is a strict local maximum. In the case that $f''(x)=0$ , being $f'''(x)\neq 0$ , $x$ is said to be an inflexion point (also called turning point). A typical example is $f(x)=\sin x$ , $f''(x)=-\sin x=0$ , $x=n\pi$ , $n=0, \pm 1, \pm 2, \dots$ , $f'''(x)=-\cos x$ , $f'''(n\pi)=-\cos n\pi=(-1)^{n+1}\neq 0$ .