Existing in Mathlib · Calculus and analysis

Mean Value Theorem

If a real-valued function is continuous on a closed interval and differentiable between its endpoints, then at some strictly interior point its instantaneous rate of change equals its average rate of change across the interval.

derivative
continuity
secant slope
interior witness

Exact theorem

Exact Mathlib statement

theorem exists_deriv_eq_slope (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfd : DifferentiableOn ℝ f (Set.Ioo a b)) : ∃ c ∈ Set.Ioo a b, deriv f c = (f b - f a) / (b - a)

The theorem at a glance

Mean Value Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory editorial poster states the Mean Value Theorem hypotheses, secant-slope formula, exact scope, and a three-movement checked route beside a curve with parallel tangent and secant lines. — Continuity and differentiability force an interior derivative to equal the endpoint secant slope.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Scientific figure illustrating: An interior tangent matches the secant between the two endpoint values. — An interior tangent matches the secant between the two endpoint values.

Why it matters

A mathematical landmark

The Mean Value Theorem turns local derivative information into a statement about finite change. It underlies fundamental estimates, monotonicity arguments, and much of one-variable calculus.

Proof architecture

How an interior derivative reaches the endpoint slope

3 curated stages

Mathlib reaches the real one-variable Mean Value Theorem through Cauchy's auxiliary construction: equalize endpoint values, invoke Rolle for an interior zero derivative, specialize the second function to the identity, and divide by the nonzero interval length.

These source-anchored stages explain the retained Mathlib proof route. They are generated explanations, not proof evidence or an extracted Lean proof-term dependency graph.

Cauchy's auxiliary folds the endpoint differences together
Cauchy's auxiliary folds the endpoint differences together
Cauchy's auxiliary built from general functions f and g has equal values at the two endpoints, preparing a Rolle witness.
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Cauchy's auxiliary built from general functions f and g has equal values at the two endpoints, preparing a Rolle witness.
h x = (g b - g a) * f x - (f b - f a) * g x and h a = h b
Mathlib proves Cauchy's mean value theorem by defining h x = (g b - g a) * f x - (f b - f a) * g x. Direct algebra gives h a = h b. For Lagrange's theorem the second function will be the identity, but the equal-endpoint construction is established first.
Rolle's witness gives the identity-specialized slope relation
Rolle's witness gives the identity-specialized slope relation
Rolle's horizontal auxiliary tangent yields the cross-multiplied relation; for the identity specialization, the tangent to f at the interior witness is parallel to the endpoint secant.
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Rolle's horizontal auxiliary tangent yields the cross-multiplied relation; for the identity specialization, the tangent to f at the interior witness is parallel to the endpoint secant.
(b - a) * f' c = (f b - f a) * 1 for some c in (a,b)
The upper auxiliary h begins and ends at equal height and is smooth at the strictly interior point c, where its gold tangent is horizontal. The lower green graph for f is one smooth cubic-spline curve through its two endpoint markers and c. Its solid gold tangent at c has exactly the same direction as the dashed secant from a to b, matching the displayed specialization (b - a) f'(c) = (f(b) - f(a)) · 1. The schematic does not choose a unique witness, allow an endpoint witness, or claim differentiability at the endpoints.
The interior derivative matches the endpoint secant
The interior derivative matches the endpoint secant
Division by the nonzero interval length and the derivative bridge turn the cross-multiplied witness into the exact selected equality.
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Division by the nonzero interval length and the derivative bridge turn the cross-multiplied witness into the exact selected equality.
∃ c ∈ (a,b), deriv f c = (f b - f a) / (b - a)
Since a<b, the interval length is nonzero, so the source rearranges the cross-multiplied equation into the secant-slope formula. exists_deriv_eq_slope then converts differentiability at the interior point to HasDerivAt with deriv f, yielding an interior c whose derivative equals the endpoint slope.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes the real one-variable Lagrange mean-value theorem: continuity on [a,b] and differentiability on (a,b) produce an interior point whose derivative equals the secant slope. It does not claim a vector-valued equality or a new proof.

Proof Atlas did not originate the Mean Value Theorem or Mathlib's declaration.
The theorem does not choose a unique witness or allow an endpoint witness.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: exists_deriv_eq_slope in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.mean-value-theorem.v001
Source: Open the pinned upstream reference