Classifying Critical Points when the 2nd derivative test FAILS

[jegues]

I'm having some trouble classifying critical points when the 2nd derivative test fails.

The function is,

$$f(x,y) = x^4 - 3x^2y^2 + y^4$$

The only critical point I've found is (0,0). How do I know that this is the only possible critical point?

Anyways, 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.

Set x=0,

z = y^4

Set y=0,

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?

Any ideas/suggestions?

 

[fzero]

All lines through the critical point are of the form $$y= ax$$. You can consider the values of $$z$$ along these lines and determine the positivity as a function of the parameter $$a$$.