Multivariable Taylor expansions and Hessians

Let f be twice continuously differentiable on an open neighborhood of a. To study a nearby point a+h, consider the one-variable function φ(t)=f(a+th) along the segment. The chain rule gives φ′(0)=∇f(a) dotted with h and φ″(t)=h transposed times Hf(a+th) times h. Applying one-variable Taylor expansion yields the quadratic model. The factor one half comes from integrating the linear change of the first derivative; it is not a special adjustment invented for multivariable notation.

For a twice continuously differentiable real function, mixed partial derivatives agree, so the Hessian is symmetric. The off-diagonal contribution occurs twice inside the quadratic form before multiplication by one half. For two coordinates, the quadratic term is one half fxx h² plus fxy hk plus one half fyy k². Forgetting that doubling produces an incorrect coefficient for cross terms. With only weaker differentiability assumptions, mixed derivatives need not agree; the regularity hypothesis should be stated rather than inferred from notation.

At a stationary point, the linear term vanishes. A positive definite Hessian makes the quadratic term positive in every nonzero direction, with a uniform positive lower bound on the unit sphere. Continuity of the Hessian ensures that the smaller remainder cannot overturn that sign sufficiently near the point, giving a strict local minimum. A negative definite Hessian gives a strict local maximum. An indefinite Hessian supplies directions of both signs and therefore a saddle. These are local conclusions and do not establish global optimality.

A positive semidefinite Hessian alone does not settle the question. The functions x⁴+y⁴ and x⁴−y⁴ have the same zero Hessian at the origin, but one has a strict minimum and the other a saddle. Higher-order terms decide the difference. In two dimensions, the determinant test is a convenient summary: positive determinant together with a positive first diagonal entry gives positive definiteness; negative determinant gives indefiniteness. A zero determinant is inconclusive. Applying this test away from a stationary point does not classify a local extremum there.

The Hessian at a single point describes leading curvature but does not bound a finite step unless surrounding curvature is controlled. The integral remainder averages Hessian values along the segment with weight 1−t. Taking an operator-norm bound M over that segment gives the displayed estimate. If third derivatives are uniformly bounded in an appropriate multilinear norm, a quadratic-model error can instead be bounded by a constant times the cube of displacement length. Each extra accuracy claim therefore comes with an extra regularity and boundedness requirement.

For a quadratic polynomial, the Hessian is constant and the second-order formula is exact. Such examples are excellent for checking signs and factors, but their zero higher-order remainder should not be generalized to nonlinear functions. In optimization, an eigenvector direction of a symmetric Hessian identifies a principal curvature, while a small eigenvalue can signal a nearly flat direction. It does not automatically indicate numerical error or a symmetry; either interpretation needs additional analysis of the function and the scale of its coordinates.