determine maxima or minima of $f(x,y)$

Question

Let $f(x,y)=x^6+2x^2y-x^4y+2y^2$, then at $(0,0)$ have maxima, minima, or saddle point.

I have check $(0,0)$ is critical point but $f_{xx}f_{yy}-f^2_{xy}=0$. How to comment on whether it's maxima or minima or saddle point at the origin?

Answer 1

Along parabolas $y=x^2, y=-x^2,y=-0.5x^2$ are the values respectively $$f(x,x^2)=x^6-2x^4-x^6+2x^4=0,$$ $$f(x,-x^2)=2x^6,$$ which is positive for the points close to $(0,0),$ $$f(x,-0.5x^2)=0.5x^4(3x^2-1),$$ negative close to $(0,0),$
therefore $(0,0)$ is a saddle point.

Answer 2

$f(x,y)=x^6+2x^2y-x^4y+2y^2$

In a neighborhood around the origin, the lowest degree terms are going to dominate. In this case, that would be the $2x^2y+2y^2$ terms. We can just analyze these terms.

If $y<0$ and $|x^2|>|y|$ then $2x^2y+2y^2 < 0$ and that is less $f(0,0).$