I am trying to find the stationary points for:

$\displaystyle f(x,y)=(x^2+y^2)e^{-(x+y)}$

So I have:

$\displaystyle f_x=0 \rightarrow -e^{-(x+y)}(y^2+x(x-2))=0$

with solutions $\displaystyle (0,0),(1,-1),(1,1),(2,0)$

$\displaystyle f_y=0 \rightarrow -e^{-(x+y)}(x^2+y(y-2))=0$

with solutions $\displaystyle (0,0),(1,1),(-1,1),(0,2)$

Do I just take the solutions that satisfy both $\displaystyle f_x$ and $\displaystyle f_y$ , i.e. $\displaystyle (0,0),(1,1)$ as the stationary points?