CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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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.

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For one regular constraint, the objective gradient is a scalar multiple of the constraint gradient

Statement

Let $f, G : U \subseteq R^{m} \to R$ be $C^{1}$ , and suppose $a$ is a local maximum or minimum of $f$ subject to $G (x) = c$ . If $\nabla G (a) \neq 0$ , then there is a unique scalar $λ$ such that $\nabla f (a) = λ \nabla G (a) .$

Facts & Assumptions

Given: The functions, constrained local extremum, and nonzero constraint gradient.

[L1]

For a scalar function, the Jacobian is the row $\nabla G (a)^{T}$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case), so $D G (a) : R^{m} \to R$ is surjective exactly when $\nabla G (a) \neq 0$ .

[L2]

At a constrained local extremum with $D G (a)$ surjective, there is a unique $λ \in R$ such that $\nabla f (a) = D G (a)^{T} λ$ (Lagrange multipliers for a regular vector-valued level-set constraint).

Proof

technique · direct

1.1

By [L1], the nonzero-gradient hypothesis makes $D G (a)$ surjective.

givenL1

2.1

Apply [L2]. Since the transpose of the one-row matrix $D G (a)$ sends $λ$ to $λ \nabla G (a)$ , its conclusion is the displayed equation.

step 1.1L2algebra

3.1

Uniqueness is part of [L2] and also follows directly from $\nabla G (a) \neq 0$ .

step 2.1∎

Depends on

Used by

Lagrange multipliers locate the extrema of a linear functional on a sphere Example

Dependency tree · two levels

12 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

University of Toronto MAT237 notes, Section 2.8 (standard reference, not scraped)