I was given the function $f(x,y,z)=4xyz-x^4-y^4-z^4$ and was asked to find the critical points and classify them using Hessian of f at the critical points. When I solved for one of the critical points it was at (0,0,0) and when I went to solve for the Hessian of f it gave me a zero matrix. Know according to Hessian,when the determinant of H at the critical point is 0, the test is inconclusive. Obviously, since the Hessian matrix is zero, the determinant must be zero as well.
My question is, is the answer just stated as inconclusive or is their another step to classify the critical points? I looked in my lecture notes and textbook and found nothing.