Page "Hessian matrix" Paragraph 19

The following test can be applied at a non-degenerate critical point x.

If the Hessian is positive definite at x, then f attains a local minimum at x.

If the Hessian is negative definite at x, then f attains a local maximum at x.

If the Hessian has both positive and negative eigenvalues then x is a saddle point for f ( this is true even if x is degenerate ).

Otherwise the test is inconclusive.