Summary: Let fx = fy = 0 at (x0, y0).
Discriminant = fxxfyy - (fxy)2 If discriminant > 0 and fxx(x0, y0) > 0, then (x0, y0, z0) is a local minimum point.
If discriminant > 0 and fxx(x0, y0) < 0, then (x0, y0, z0) is a local maximum point.
If discriminant < 0, then (x0, y0, z0) is a saddle point.
If discriminant = 0, the 2nd derivative test is inconclusive.