For the function $f(x,y)=x^2y^2 + \frac{1}{y} + \ln(\frac{1}{x})$ I get two critical points, namely $P_1 =\left(\sqrt{\frac{1}{2}},1 \right)$ and $P_2 =\left(-\sqrt{\frac{1}{2}},1 \right)$. However the domain of my function is $Df=\{(x,y) \in \mathbb{R}^2; x > 0, y \neq 0\}$.
Now point $P_2$ is not in the domain of function$f$ and I think it should be immediately ruled out, but the Hessian matrix shows that this point is a minimum. Should it be ruled out or not?