94 - The Second Partial Derivative Test - Making Sense of What's What
When we have a stationary point, we have a choice to determine whether this point is a minimum, maximum or saddle point. Much like the second derivative test in single-variate calculus, the second partial derivative test applies here. And to do this, we use the the determinant of the hessian.
Let
\begin{cases}
H > 0 & \text{The point is either a maximum or minimum}
\
H = 0 & \text{The test is useless}
\
H < 0 & \text{The point is a saddle point}
\end{cases}