I'm having some trouble classifying critical points when the 2nd derivative test fails.
The function is,
The only critical point I've found is (0,0).
The second derivative test fails for this point, so I've got to classify it another way.
I can try to get a rough idea of what the surface looks like by drawing cross sections, and maybe I can get a good enough picture to classify my point.
z = y^4
Z = x^4
So at the bottom sits my critical point and I have to parabolas in the y-z plane and x-z plane.
This makes me think my point may be a relative min.
How can I figure out whether all values of Z are positive or not?
Your partial derivatives are fine. My mistake!
Since your second derivative test fails, you might want to see what happens to some paths containing . You can show that taking the path gives a function increasing away from . On the other hand, taking the path gives a function decreasing away from . Therefore, you must be on a saddle.
I actually had the benefit of a computer-generated contour plot to see which paths to take. If you were to do this purely by hand, you would probably have to stare at your function and toy with different paths and see what comes out. I would imagine that the situation becomes much more difficult when you are actually sitting on a maximum or minimum.