Answer
$f_x(x,y)=0$ and $f_y(x,y)=0$ simultaneously for $x=0$ and $y=0.$
Work Step by Step
First we will find both partial derivatives. The partial derivative with respect to $x$ is:
$$f_x(x,y)=\frac{\partial}{\partial x}(x^2-xy+y^2)=2x-y$$
The partial derivative with respect to $y$ is:
$$f_y(x,y)=\frac{\partial}{\partial y}(x^2-xy+y^2)=-x+2y$$
To find all values of $x$ and $y$ such that $f_x(x,y)=0$ and $f_y(x,y)=0$ we will equate both partial derivatives with $0$:
$$f_x(x,y)=2x-y=0,f_y(x,y)=2y-x=0$$
From the first equation we have: $y=2x$. Putting this into the second equation we get:
$$2\cdot2x-x=0\Rightarrow3x=0\Rightarrow x=0,$$
which gives us $y=2x=2\cdot0=0$
So, $f_x(x,y)=0$ and $f_y(x,y)=0$ simultaneously for $x=0$ and $y=0.$
