TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The multivariable second-derivative test by definiteness of the Hessian

Statement

Let $a$ be a critical point of a $C^2$ scalar field. A positive definite Hessian gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and an indefinite Hessian gives neither. If the Hessian is semidefinite but not definite, the Hessian test gives no conclusion in general.

Facts & Assumptions

Given: A $C^2$ scalar field and a critical point $a$ .

[L1]

The second-order Taylor expansion has quadratic term $\tfrac12\langle H_f(a)h,h\rangle$ and remainder $o(\|h\|^2)$ (Second-order Taylor expansion $f(a+h)=f(a)+\nabla f(a)\cdot h+\tfrac12h^TH_f(a)h+o(\|h\|^2)$ ).

[L2]

A definite quadratic form has a uniform signed bound on the unit sphere (A definite quadratic form has a uniform signed bound on the Euclidean unit sphere).

Proof

technique · direct

1.1

At a critical point the linear term in [L1] vanishes.

L1given

2.1

Write $h=ru$ with $r=\|h\|_2$ and $\|u\|_2=1$ when $h\ne0$ . The sign bound in [L2] dominates the $o(r^2)$ remainder for sufficiently small $r$ .

step 1.1L2algebra

3.1

This gives the strict minimum and maximum conclusions in the definite cases; two unit directions of opposite quadratic sign give neither extremum in the indefinite case.

step 2.1algebra

4.1

For a semidefinite Hessian which is not definite, its quadratic form has a nonzero null direction, so step 2.1 supplies no signed quadratic bound. No universal conclusion is possible: at $0$ the one-variable functions $x^4$ , $-x^4$ , and $x^3$ all have zero Hessian, but respectively have a strict local minimum, a strict local maximum, and neither.

step 2.1algebra∎

Depends on

Used by

The two-variable Hessian determinant test Corollary
A zero Hessian occurs at a strict minimum, a strict maximum, and a saddle Counterexample
x²+y²(1-x)³ has a unique critical point, a strict local but nonglobal minimum Counterexample
The monkey saddle x³-3xy² has an indefinite higher-order critical point Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Analysis, Convexity, and Optimization (standard reference, not scraped)