LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Hessian is negative semidefinite at an interior local maximum

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $f \in C^{2} (U)$ , and let $a \in U$ be a local maximum of $f$ . Then $h^{T} H_{f} (a) h \leq 0 for every h \in R^{n};$ in particular $\nabla f (a) = 0$ and $Δ f (a) = tr H_{f} (a) \leq 0$ .

Facts & Assumptions

Given: An open $U \subseteq R^{n}$ , $f \in C^{2} (U)$ , an interior local maximum $a \in U$ , and an arbitrary $h \in R^{n}$ .

[F1]

At an interior local extremum of a differentiable scalar field the gradient vanishes: $\nabla f (a) = 0$ (Fermat's theorem: an interior differentiable local extremum has zero gradient).

[F2]

For a $C^{2}$ scalar field, $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} ⟨ H_{f} (a) h, h ⟩ + o (∥ h ∥^{2})$ (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ ).

[F3]

The Hessian $H_{f} (a)$ is the matrix of second partial derivatives, and $Δ f = \sum_{i} \partial_{i} \partial_{i} f$ is the trace of the Hessian (The Hessian matrix and critical points of a scalar field, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F4]

Local maximality means $f (a) \geq f (x)$ for all $x$ in some Euclidean neighbourhood of $a$ (Local and strict local extrema for scalar fields on Euclidean open sets).

Proof

Given: An open $U \subseteq R^{n}$ , $f \in C^{2} (U)$ , an interior local maximum $a \in U$ , and $h \in R^{n}$ .

1.1F1F4given

Since $a$ is an interior point of $U$ at which $f$ has a local maximum, [F1] applies and gives $\nabla f (a) = 0$ .

2.1step 1.1F2F4given

Suppose $h^{T} H_{f} (a) h = : c > 0$ ; then $h \neq 0$ and [F2] and step 1.1 give $f (a + t h) = f (a) + \frac{t^{2}}{2} h^{T} H_{f} (a) h + o (t^{2}) = f (a) + t^{2} (\frac{c}{2} + o (1)) > f (a)$ for every sufficiently small $t \neq 0$ , and $a + t h$ lies in the neighbourhood of $a$ on which the local maximum is attained for small $t$ ; this contradicts [F4]. Hence $h^{T} H_{f} (a) h \leq 0$ , and since $h$ was arbitrary the quadratic form of the Hessian is negative semidefinite.

3.1step 2.1F3given∎

Taking $h = e_{i}$ in step 2.1 gives $H_{i i} (a) \leq 0$ for every coordinate index $i$ , and by the trace formula [F3] the Laplacian is $Δ f (a) = \sum_{i} H_{i i} (a) \leq 0$ ; together with step 1.1 this is the stated conclusion.

Depends on

Used by

Strict-subsolution perturbation for the heat operator Lemma

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext, Springer 2011) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)