relations between Hessian matrix and local extrema

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

If $H(x)$ is positive definite (http://planetmath.org/PositiveDefinite), 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 (http://planetmath.org/NegativeDefinite), 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. $f:ℝ\to ℝ$), 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 $f^{\prime }(x)=0$. If $f^{\prime \prime }(x)>0$, then $x$ is a strict local minimum. If , then $x$ is a strict local maximum. In the case that $f^{\prime \prime }(x)=0$, being $f^{\prime \prime \prime }(x)\ne 0$, $x$ is said to be an inflexion point (also called turning point). A typical example is $f(x)=\sin x$, $f^{\prime \prime }(x)=-\sin x=0$, $x=n\pi$, $n=0,\pm 1,\pm 2,\mathrm{\dots }$,$f^{\prime \prime \prime }(x)=-\cos x$, $f^{\prime \prime \prime }(n\pi )=-\cos n\pi ={(-1)}^{n+1}\ne 0$.