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Can anyone explain what a positive determinant of the Hessian matrix of second order partial derivatives at a certain point and a 0 minor determinant of the matrix corresponds to? I know the other cases, for example both being positive corresponds to being a local minimum and so on. Does such a case mean the second derivative test is inconclusive?

Thank you

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The determinant of the 2-variable Hessian corresponds to the 2nd derivative of a single-variable function. In both cases, if the quantity is positive, then the function is convex at that point.

If the dimension is $>2$, the convexity condition on the Hessian (on the matrix itself, not on its determinant) is that it must be positive-definite.

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