4.15 Example of discriminant test

Solution. We find the stationary points, then use the Theorem to classify them. To start with, we calculate the partial derivatives up to second order.

We already saw in 4.3 that $f_{x}=3x^{2}-3$ and $f_{y}=3y^{2}-12$. Thus $f_{xx}=\,{6x,}$ $f_{xy}=\,{0}$ and $f_{yy}=\,{6y.}$ It follows that the Hessian discriminant is $f_{xx}f_{yy}-f_{xy}^{2}=\,{36xy.}$

In particular, $\Delta_{P}$ is positive if $x$ and $y$ have the same sign, and is negative if they have opposite signs. Moreover, $f_{xx}$ is positive if and only if $x>0$.