Alphabeta Math
TheoremStatement: 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.

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Fermat's theorem: an interior differentiable local extremum has zero gradient

Statement

If f:U⊆Rm→R is differentiable at an interior local maximum or minimum a, then ∇f(a)=0.

Facts & Assumptions

Given: A differentiable scalar field with a local extremum at a.

[L1]

The one-variable Fermat theorem gives derivative zero at an interior differentiable local extremum (Fermat's interior extremum theorem: if f has a local extremum at a point c interior to its domain and is differentiable at c, then f′(c)=0).

[L2]

The gradient consists of the coordinate partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Proof

technique · direct
1.1

Restrict f to each coordinate line through a. The restriction has a local extremum at 0, so [L1] makes its derivative zero.

L1given
2.1

These derivatives are the entries of ∇f(a) by [L2], so every entry vanishes.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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