For a 2 by 2 Hessian, simply use the criteria in the second partial derivative test:negative detrminant gives a saddle point, positive determinant and positive second derivative with respect to x a local minimum, and positive determinant and positive second derivative with respect to x a local maximum.
For 3 by 3 Hessian, if it is positive-definite (or equivalently xTHx >0 for all non-zero real vectors x where xT is transpose of x) then it attains an isolated local minimum. If the Hessian is negative-definite, then it attains an isolated local maximum. If the Hessian has both positive and negative eigenvalues then it has a saddle point. Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. Also remember these should be calculated at a given point of interest (x,y,z).
