 Solution

At an interior stationary point, a negative determinant of the symmetric Hessian matrix means its two eigenvalues have opposite signs. Along their eigenvectors the leading quadratic change has opposite signs, and the higher-order remainder cannot reverse these signs sufficiently close to the point. Hence the nondegenerate saddle condition isWhen the determinant vanishes, a saddle is still possible but is not determined by second derivatives alone: $x^4-y^4$ is a simple example. Thus these tests are decisive for nondegenerate stationary points, not an exhaustive classification from first and second derivatives of every smooth function.