Critical points of $f(x,y)=x^2+xy+y^2+\frac{1}{x}+\frac{1}{y}$ I would like some help finding the critical points of $f(x,y)=x^2+xy+y^2+\frac{1}{x}+\frac{1}{y}$. I tried solving $f_x=0, f_y=0$ (where $f_x, f_y$ are the partial derivatives) but the resulting equation is very complex. The exercise has a hint: think of $f_x-f_y$ and $f_x+f_y$. However, I can't see where to use it.
Thanks!

Here $f_{x}=2x+y-\frac{1}{x^{2}}$ and $f_{y}=2y+x-\frac{1}{y^{2}}$
For critical point
$$f_{x}=0,\ f_{y}=0$$
$$2x+y-\frac{1}{x^{2}}=0\ ,\ 2y+x-\frac{1}{y^{2}}=0$$ 
$$2x^{3}+x^{2}y-1=0\ ,\ 2y^{3}+xy^{2}-1=0$$
Substracting this two equations we get,
$$ 2(x^{3}-y^{3})+xy(x-y)=0$$
$$ 2(x-y)(x^{2}+xy+y^{2})+xy(x-y)=0$$
$$(x-y)(2x^{2}+2xy+2y^{2}+xy)=0$$
$$x-y=0\ \Rightarrow x=y$$
Put $x=y$ in $2x^{2}+2xy+2y^{2}+xy=0$,we get,
$$ 2x^{2}+2x^{2}+2x^{2}+x^{2}=0\ \Rightarrow x=0 $$
Also $y=0$ \
$\therefore (0,0)$ is a critical point.
