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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?

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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.

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