Second derivative test for a function of two variables

Statement

Suppose $f$ is a function of two variables $x,y$. Suppose $(x_0,y_0)$ is a point in the domain of $f$ such that both the first-order partial derivatives at the point are zero, i.e., $f_x(x_0,y_0)=f_y(x_0,y_0)=0$. Suppose that all the second-order partial derivatives (pure and mixed) for $f$ exist and are continuous at and around $(x_0,y_0)$. Note that by Clairaut's theorem on equality of mixed partials, this implies that $f_{xy}(x_0,y_0)=f_{yx}(x_0,y_0)$.

The second derivative test helps us determine whether $f$ has a local maximum at $(x_0,y_0)$, a local minimum at $(x_0,y_0)$, or a saddle point at $(x_0,y_0)$.

First, consider the Hessian determinant of $f$ at $(x_0,y_0)$, which we define as:

$D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-(f_{xy}(x_0,y_0){)}^2$

Note that this is the determinant of the Hessian matrix:

$H(f)(x_0,y_0)=\left(\begin{array}{cc}{f}_{xx}(x_0,y_0)& f_{xy}(x_0,y_0)\\ f_{yx}(x_0,y_0)& f_{yy}(x_0,y_0)\\ \end{array}\right)$

We now have the following:

Case	Local maximum, local minimum, saddle point, or none of these?
$D<0$	Saddle point
$D>0$ and $f_{xx}(x_0,y_0)>0$	Local minimum (reasoning similar to the single-variable second derivative test)
$D>0$ and $f_{xx}(x_0,y_0)<0$	Local maximum (reasoning similar to the single-variable second derivative test)
$D=0$ and one or both of $f_{xx}(x_0,y_0)$ and $f_{yy}(x_0,y_0)$ is positive	Inconclusive, but we can rule out the possibility of being a local maximum
$D=0$ and one or both of $f_{xx}(x_0,y_0)$ and $f_{yy}(x_0,y_0)$ is negative	Inconclusive, but we can rule out the possibility of being a local minimum
All entries of the Hessian matrix are zero, i.e., $f_{xx}(x_0,y_0),f_{yy}(x_0,y_0),f_{xy}(x_0,y_0)$ are all zero	Inconclusive. No possibility can be ruled out.