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I have solved for the critical points using the gradient, and I have solved for the Hession, which yields a 2x2 matrix. I have plugged in my critical points into the gradient.

Now, do I apply the same rules as in linear algebra where I find the determinant and trace to calculate positive definite, negative definite, and indefiinite? I currently add up the diagonal and check the determinant. If they are both positive, then it's a local min, if they are both negative, it's a local max, and a saddle point if none of the above, correct?