I have to find the critical points of the function f and to tell whether they are local minimum, local maximum or saddle point.

The function is $\displaystyle f(x,y)=x^2y^2$.

By intuition the point (0,0) is a local minimum.

Formally, I found that $\displaystyle f_x(x,y)=0 \Leftrightarrow 2xy^2=0$, so $\displaystyle x=0$ and $\displaystyle y=$ any value. (I'm not convinced since y is treated as a constant and it might be 0, while in this case x could be any other number different from 0. Hence my "so" is not an implication in fact! I don't understand what I'm doing wrong)

$\displaystyle f_y(x,y)=0 \Rightarrow y=0$ and $\displaystyle x=$ any value. (Once again I don't trust my implication)

Hence the critical point is $\displaystyle (0,0)$.

From my class notes, if $\displaystyle f_{xx}(0,0)f_{yy}(0,0)-f_{xxy}(0,0)=0$, then I cannot conclude about the nature of the critical point. Unfortunately it happens here, and there's no theorem that can help me.

If it was the case of a transcendantal function, I would have approximate it by a Taylor's polynomial and checking if the polynomial reaches a minimum or a maximum or a saddle point, but the problem is that the function is a polynomial. (by the way I'm not even sure that approximating a function via a Taylor's polynomial assegurate that the approximated function reaches the same critical points. Maybe there's a theorem about it but I never read it yet).