I am trying to find and classify the critical points of:
$$f(x,y)=\frac{x^3+y}{3x^2+y^2}$$
so far I have found the partial derivatives:
$$\frac{\partial f}{\partial x}=\frac{3x^4+3y^2x^2-6xy}{(3x^2+y^2)^2}$$ $$\frac{\partial f}{\partial y}=\frac{3x^2-2yx^3-y^2}{(3x^2+y^2)^2}$$
Setting them $=0$, $(1,1)$ and $(-1,-1)$ are obviously solutions with $(0,0)$ not existing. I can't see how to solve the system though.
Firstly, are there any solutions I've missed?
Secondly, what is the best way to go about classifying them?