Second Derivative Test

In monovarietal functions, determining the nature of a Stationary Point was via the second derivative $\frac{d^{2}y}{d{x}^2}$. For multivariate functions, a matrix known as the Hessian Matrix#tosee is used.

Hessian Matrix §

Ripped straight from Wikipedia: https://en.wikipedia.org/wiki/Hessian_matrix

Bivariate Functions §

In a bivariate function $f(x,y)$, (assuming f is continuous) then $f_{xy}=f_{yx}$ (see theorem), the Hessian Matrix is as follows:

$$\left[\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{yx}& f_{yy}\end{array}\right]=\left[\begin{array}{cc}\frac{{\partial }^{2}f}{\partial x^2}& \frac{{\partial }^{2}f}{\partial y\partial x}\\ \frac{{\partial }^{2}f}{\partial x\partial y}& \frac{{\partial }^{2}f}{\partial y^2}\end{array}\right]$$

Let $H$ be the Hessian Matrix and $P=(x_0,y_0)$ be the point to be evaluated.

Then $\det (H)=f_{xx}{f}_{yy}-(f_{xy}{)}^2$

1. If $\det (H)>0$ AND $f_{xx}>0$ at $P$ : $P$ is a local minimum
2. Similarly if $\det (H)>0$ AND $f_{xx}<0$ at $P$ : $P$ is a local maximum
3. If $\det (H)<0$, $P$ is a saddle point (3D turning point.)

Warning

The second derivative test is inconclusive if $\det (H)=0$