I've got the following equation for all $(x,y)$ in $\mathbb{R}^2$ where $a$ is a real number:
$$f(x,y) = 4ay^2-x^2y^3-x^2$$
I want to calculate all critical points when:
- $a=0$
- $a>0$
- $a<0$
If I graph the various functions, I can see that with both $a<0$ and $a>0$, the function only has one critical point at the origin.
But if $a=0$ the function has critical points all along the y-axis.
Does anybody have any tips on how I can explain this in writing?