I recently learned about the Second-Derivative Test for Multivar Calc, which basically uses the Hessian Matrix at a given Critical Point.

The test states that a saddle occurs when the determinant of the matrix of 2nd derivatives is less than 0, and an extremum occurs when the determinant is greater than 0.

I understand the logic of critical points in single variable, but could somebody explain WHY this test works? ;o

Thanks!