Very weird question relating to critical points

I've done some work on this, though either I've hit a strange answer or I did something wrong. The question has two parts.

**a)** What happens to $\displaystyle D=f_{xx}f_{yy}-(f_{xy})^2$ at $\displaystyle (0,0)$ for $\displaystyle f(x,y)=x^4-2x^2y^2+y^4$? Classify the critical point at $\displaystyle (0,0)$.

Working out the problem, I got to here:

$\displaystyle f_{xx}=12x^2-4y^2$, $\displaystyle f_{xy}=-8xy$, $\displaystyle f_{xx}=12y^2-4x^2$

$\displaystyle D=-48x^4+96x^2y^2-48y^2$

This, however, leads to the answer $\displaystyle D(0,0)=0$, which is an inconclusive answer (it's not a maximum, minimum, or saddle point).

Is this the answer that's being asked of, or did I miss a detail?

**b)** Repeat for $\displaystyle f(x,y)=(y-x^3)(y-x^5)$.

I haven't gotten to this yet, but it probably won't take too long.