Well, I've gotten the first half out of the way, though the second half again has the situation where (my work is below)

$\displaystyle D=-25x^8+82x^6-9x^4-40x^3y-12xy$

$\displaystyle D(0,0)=0, f_{xx}(0,0)=0$

so again we cannot draw a conclusion from this about the critical point at $\displaystyle (0,0)$

According to a source (found

here), each factor (he wasn't very clear on what he meant by factor) has a saddle point at $\displaystyle (0,0)$.

He claims, however, that $\displaystyle f$ has a minimum at $\displaystyle (0,0)$.

Would he be right? And if so, how exactly am I supposed to show it?