Finding critical points of a multivariable function Let $f(x,y)=e^{x^2-xy+y^2}$
(a) Find all the critical points of the following function.
(b) Find the all the local maxima and local minima  of the function if there is any.
What i tried.
I tried to differentiate the following function wrt $x$ and $y$, and then equating them to $0$ ie $$(2x-y)e^{x^2-xy+y^2}=0$$ and
$$(2y-x)e^{x^2-xy+y^2}=0$$
While i could differentiate the function, the diffculty lies in solving the following equations to get $x$ and $y$. I only managed to get a critical point of $(0,0)$ but im sure there are other critical points as well .Could anyone explain.Thanks

Notice that $e^{x^2-xy+y^2} > 0$ for every pair $(x,y)$. Then what you found boils down to $$\begin{cases} 2x - y = 0 \\ 2y - x = 0 \end{cases} ,$$ which has as the only solution $(0,0)$. For the second part, look at ${\rm Hess} \ f_{(0,0)}$, and see if is is positive/negative definite, etc.
