You don't even need the eigenvalues.

Find the grad and solve it equal to the zero matrix for and .

.

Then find the Hessian:

and evaluate it at the critical points you found with the grad.

If the Hessian is positive definite, you have a minimum, if the Hessian is negative definite, you have a maximum.

(It's the equivalent of the second derivative test for a function of one variable).

You could check the eigenvalues though to check that it is a saddle point. It's only a saddle point if the Hessian has BOTH negative and positive eigenvalues.