I was trying to find the critical points of $f:\{ (x,y) \in \mathbb{R^2} \mid xy \neq 0\}\to \mathbb{R}$ given by $f(x,y) = x^2 + y^2 + \frac{1}{xy}$
Then I made $ \nabla f(x,y)=(0,0) $ and obtained the points $P_1 = (2^{-1/4},2^{-1/4})$ and $P_2=(-2^{-1/4},-2^{-1/4})$
But when I looked at the graph of $f$,
I saw that the points I obtained indeed look like local extremal points, but the points $P_3 = (2^{-1/4},-2^{-1/4})$ and $P_4 = (-2^{-1/4},2^{-1/4})$ are points that look like saddle points, but if they were saddle points, then $\nabla f(P_3)$ and $\nabla f(P_4)$ should be $(0,0)$, which doesn't happen. I want to understand why they look like this and if they have a special meaning for the function $f$.