# Is a negative Hessian (of a 1-D function of 2 variables) at some point sufficient to indicate a saddle point?

Is a negative Hessian (of a 1-D function of 2 variables) at some point sufficient to indicate a saddle point, or do you separately need to know that the point is a critical point?

Take $$f(x,y) = x^2y^2$$. Then $$\det(Hf) = -12x^2y^2$$. This is negative for all $$\{(x, y) \in \mathbb{R}^2: xy \neq 0\}$$ and clearly these points are not all saddle points.