Find the critical points

Question

Find the critical points of $f(x,y) = 2y-3x-3xy+x^2+y^2-6.$

Every time I try this I get $(0,-1)$ as a critical point however, when I go to graph this there is no min or max??Any idea?

Answer 1

That's natural. Your computations are correct: the gradient of your function is $0$ at $(0,-1)$ and only at that point. But the Hessian of $f$ there is $\left[\begin{smallmatrix}2&-3\\-3&2\end{smallmatrix}\right]$, whose determinant is negative. So, $(0,-1)$ is a saddle point of $f$.