The critical points of h(x,y) occur wherever its partial derivatives
and
vanish simultaneously. We have

Substitute y in the second equation and solve for x, then for y :

This is to say there are two critical points,

To classify these critical points, we carry out the second partial derivative test. h(x,y) has Hessian

whose determinant is
. Now,
• if the Hessian determinant is negative at a given critical point, then you have a saddle point
• if both the determinant and
are positive at the point, then it's a local minimum
• if the determinant is positive and
is negative, then it's a local maximum
• otherwise the test fails
We have

while

So, we end up with
