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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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Fermat's theorem: an interior differentiable local extremum has zero gradient

Statement

If f:U⊆Rm→Rf:U\subseteq\mathbb R^m\to\mathbb R is differentiable at an interior local maximum or minimum aa, then ∇f(a)=0\nabla f(a)=0.

Facts & Assumptions

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

[L1]

The one-variable Fermat theorem gives derivative zero at an interior differentiable local extremum (Fermat's interior extremum theorem: if ff has a local extremum at a point cc interior to its domain and is differentiable at cc, then f′(c)=0f'(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 ff to each coordinate line through aa. The restriction has a local extremum at 00, so [L1] makes its derivative zero.

L1given
2.1

These derivatives are the entries of ∇f(a)\nabla 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 20 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