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 $f_{x y} (x_{0}, y_{0}) = f_{y x} (x_{0}, y_{0})$ .

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:

$D = f_{x x} (x_{0}, y_{0}) f_{y y} (x_{0}, y_{0}) - (f_{x y} (x_{0}, y_{0}))^{2}$

Note that this is the determinant of the Hessian matrix:

$H (f) (x_{0}, y_{0}) = (\begin{matrix} f_{x x} (x_{0}, y_{0}) & f_{x y} (x_{0}, y_{0}) \\ f_{y x} (x_{0}, y_{0}) & f_{y y} (x_{0}, y_{0}) \end{matrix})$

We now have the following:

Case	Local maximum, local minimum, saddle point, or none of these?
$D < 0$	Saddle point
$D > 0$ and $f_{x x} (x_{0}, y_{0}) > 0$ (note that these together also force $f_{y y} (x_{0}, y_{0}) > 0$ )	Local minimum (reasoning similar to the single-variable second derivative test)
$D > 0$ and $f_{x x} (x_{0}, y_{0}) < 0$ (note that these together also force $f_{y y} (x_{0}, y_{0}) < 0$ )	Local maximum (reasoning similar to the single-variable second derivative test)
$D = 0$ and one or both of $f_{x x} (x_{0}, y_{0})$ and $f_{y y} (x_{0}, y_{0})$ is positive (note that if one of them is positive, the other one is either positive or zero)	Inconclusive, but we can rule out the possibility of being a local maximum
$D = 0$ and one or both of $f_{x x} (x_{0}, y_{0})$ and $f_{y y} (x_{0}, y_{0})$ is negative (note that if one of them is negative, the other one is either negative or zero)	Inconclusive, but we can rule out the possibility of being a local minimum
All entries of the Hessian matrix are zero, i.e., $f_{x x} (x_{0}, y_{0}), f_{y y} (x_{0}, y_{0}), f_{x y} (x_{0}, y_{0})$ are all zero	Inconclusive. No possibility can be ruled out.