$x^3-3xy^2$ Show critical point is saddle

Question

So far I have managed to get critical point is at $(0,0,0)$ and that second derivative test fails at the critical point. What test can be done to show that this critical point is a saddle point?

Answer 1

HINT: write the hessian matrix of your function and observe that is not semipositive neither seminegative defined. This mean there exist a direction whose eigenvalue is positive and another direction whose eigenvalue is negative, then the graph is a saddle since the graph is not convex neither concave.

Answer 2

Differentiate twice, and do it with respect to first x and then y. One result should be negative and the other positive, which means that the point is a maximum in one direction and a minimum in the other, which is the definition of a saddle point.

If it fails at that point, check for errors in your work and assuming none, manually check the function immediately either side of the critical point to see if it's greater or smaller.

Sketch it to gain insight.