The number of critical points of the function $f(x,y)=(x^2+3y^2)e^{-(x^2+y^2)}$ is 
The number of critical points of the function $$f(x,y)=(x^2+3y^2)e^{-(x^2+y^2)}$$ is _____.

My attempt:
$f_x=-(2x^3+(6y^2-2)x)e^{-(x^2+y^2)}$ and $f_y=-(6y^3+(2x^2-6)y)e^{-(x^2+y^2)}$
Clearly, $(0,0)$ is a critical point since at $(0,0)$ we have: $f_x=0, f_y=0$. Are there anymore, if yes how to find them?

Clearly, you can factor:
\begin{align}
f'x&=-2x(x^2+3y^2-1)\mathrm e^{-(x^2+y^2)} \\
f'y&=-2y(x^2+3y^2-3)\mathrm e^{-(x^2+y^2)}
\end{align}
and observe that $x^2+3y^2-1$ and $x^2+3y^2-3$ can also be $0$, but not simultaneously. Therefore you also have the solutions of the systems
$$
\begin{cases}
x=0\\x^2+3y^2-3=0
\end{cases}\quad\text{ and }\quad
\begin{cases}
y=0\\x^2+3y^2-1=0
\end{cases}
$$
