I must find and classify all the critical points in the following function: $$ f(x,y)= x^2 + y^2 +x^2y +4$$ I have said that $$f_x=2x+2xy=0$$ $$ 2x = -2xy$$ $$ \frac{ 2x}{\ -2x}=y $$ $$y=-1$$ $$f_x = 2y +x^2 =0 $$ $$ -2 +x^2 = 0$$ $$x=±\sqrt{2}$$
Therefore my critical points are $$(±\sqrt{2}, -1) $$
Is this right? I only have two critical points and both can be classified as saddle points so I am missing a max and a min.