# Extrema in two variable equations

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

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.