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.

The Second Partial Derivative Test

Let be the determinant of the hessian, when , for the hessian matrix Then, depending on the sign of the determinant, we have three scenarios,$$
\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}

In the case of a two-input function $f(x,y)$ we can use a much more convenient formula for the determinant of the hessian,$$H_{det} = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-(f_{xy}(x_0,y_0))^2$$ # [[95 - The Line Integral - Arc Length]]