For a twice continuously differentiable function, a finite second-order Taylor expansion about an interior point iswhere , , at that point. A differentiable interior extremum requires . For a strict minimum guaranteed by the quadratic term, the Hessian matrix must be positive definite. Completing the square gives . Thus sufficient nondegenerate minimum conditions areIf , the second-order test is inconclusive; for example has a strict minimum at zero despite a zero Hessian. The finite Taylor argument only needs regularity. The source's infinite Taylor representation additionally needs suitable analyticity: a smooth flat function, such as extended by zero at the origin, has every Taylor coefficient zero there but is positive nearby.
At an interior stationary point, let , , and . The quadratic change in the Taylor expansion is negative in every nonzero displacement when the Hessian matrix is negative definite. Applying the minimum test to gives the sufficient nondegenerate maximum conditionsThe quadratic term then dominates the smaller Taylor remainder in a sufficiently small neighborhood, proving a strict local maximum. A vanishing determinant requires higher-order analysis instead.
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: 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.
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