Find and classify the stationary points:

$\displaystyle f(x,y)=xy+x^{-1}+y^{-2}$

$\displaystyle f_x=y-x^{-2}$

$\displaystyle f_y=x-y^{-2}$

$\displaystyle f_x=0\Rightarrow y-\frac{1}{x^2}=0$

$\displaystyle f_y=0\Rightarrow x-\frac{1}{y^2}=0$

Is $\displaystyle (1,1)$ the only stationary point?

Can $\displaystyle (0,0)$ be a stationary point?