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!

You should solve $f_{x}=0,f_{y}=0$ and we have $2x+y-1/x^{2}=0,2y+x-1/y^{2}=0$. They are just $2x^{3}+x^{2}y-1=0, 2y^{3}+y^{2}x-1=0$. Consider symmetric solutions $(a,a)$ we have $$2a^{3}+a^{3}-1=0\leftrightarrow a^{3}=\frac{1}{3}\leftrightarrow x=y=3^{-\frac{1}{3}}$$
For the non-symmetric case, Solve $2x^{2}+2y^{2}+3xy=0$ implies no real solutions other than $x=y=0$, as the determinant is $9y^{2}-4*2*2y^{2}=-7y^{2}<0$ for $y\not=0$.
This ignores the original point $(0,0)$, which is not defined. 
