Example 3 find the local maximum and minimum values and saddle points of f(x, y) = x4 + y4 − 4xy + 1. solution we first locate the critical points:

We have critical points whenever both partial derivatives vanish:





The Hessian is




At (0,0), we have , so (0,0) is a saddle point.

At (-1, -1), we have and , so (-1, -1) is a local minimum.

At (1, 1), we have and , so (1, 1) is also a local minimum.