CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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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.

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

Lagrange multipliers for a regular graph constraint $y = ψ (x)$

Statement

Let $V \subseteq R^{m}$ and $W \subseteq R^{m + n}$ be open, let $x_{0} \in V$ , let $ψ : V \to R^{n}$ be differentiable at $x_{0}$ , put $a = (x_{0}, ψ (x_{0})) \in W$ , and let $f : W \to R$ be differentiable at $a$ . If $f ∣_{W \cap graph (ψ)}$ has a local extremum at $a$ , then for $G : V \times R^{n} \to R^{n}$ given by $G (x, y) = y - ψ (x)$ there is $λ \in R^{n}$ such that $\nabla f (a) = D G (a)^{T} λ$ .

Facts & Assumptions

Given: The hypotheses of the statement.

[L1]

A constrained local extremum annihilates every tangent velocity of a differentiable parametrization (A constrained local extremum annihilates every velocity of a differentiable parametrization).

[L2]

The gradient represents the derivative and the Jacobian records the derivative in coordinates (For a differentiable scalar field, $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ and the unit direction of steepest ascent is the normalized gradient, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Proof

technique · direct

1.1

Parametrize the graph by $Γ (x) = (x, ψ (x))$ . Since $ψ$ is differentiable at $x_{0}$ , $Γ$ is continuous there; because $V$ and $W$ are open, for every $v \in R^{m}$ the curve $t \mapsto Γ (x_{0} + t v)$ is defined and lies in $W$ for sufficiently small $t$ . Apply [L1] to get $D f (a) (v, D ψ (x_{0}) v) = 0$ .

L1givenalgebra

2.1

In block gradient coordinates, step 1.1 says $\nabla_{x} f (a) + D ψ (x_{0})^{T} \nabla_{y} f (a) = 0$ .

step 1.1L2algebra

3.1

Set $λ = \nabla_{y} f (a)$ . Since $D G (a) = (- D ψ (x_{0}), I_{n})$ , step 2.1 yields $D G (a)^{T} λ = (\nabla_{x} f (a), \nabla_{y} f (a)) = \nabla f (a)$ .

step 2.1L2algebra∎

Depends on

Used by

A constrained extremum on an affine graph satisfies the graph Lagrange rule Example

Dependency tree · two levels

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Sources

Lagrange Multipliers (standard reference, not scraped)

Lagrange multipliers for a regular graph constraint y=ψ(x)y=ψ(x)