{ "artifactId": "artifact.library.mathlib.mean-value-theorem.v001", "candidateOnly": false, "category": "Calculus and analysis", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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)", "family": "Differential calculus", "id": "library.mathlib.mean-value-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/mean-value-theorem/", "entryData": "/data/library-theorems/mean-value-theorem.json", "evidence": "/proofs/artifact.library.mathlib.mean-value-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.mean-value-theorem.v001.evidence.json", "source": "/sources/upstream/mean-value-theorem/" }, "landmark": { "completeFormalType": "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)", "concepts": [ "derivative", "continuity", "secant slope", "interior witness" ], "plainLanguage": "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.", "poster": { "alt": "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.", "byteSize": 2245663, "caption": "Continuity and differentiability force an interior derivative to equal the endpoint secant slope.", "description": "The poster presents the closed-interval and open-interval hypotheses, the exact real-valued slope equality, the Cauchy mean-value route used by the selected source, and the theorem's nonclaims.", "derivatives": [ { "byteSize": 98952, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-poster-v2-640.webp", "sha256": "sha256:04da5ba270aca592eaa69194eb63c3ff4d0dfa843d23784fe232ee87b2b59103", "width": 640 }, { "byteSize": 185978, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-poster-v2-1024.webp", "sha256": "sha256:5b3d1cc5783c18f15182814c4c54285295462a28abcaaed1f960c16bfd5df532", "width": 1024 } ], "generatedAt": "2026-07-22T05:30:00Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-poster-v2.png", "sha256": "sha256:9a6f51d36b649bf437f82dfa4b6ec63d618d79ca18463e0e9a1522a6642d1ced", "title": "Mean Value Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "REAL ANALYSIS · DIFFERENTIAL CALCULUS\nMEAN VALUE THEOREM\na < b\nf is continuous on [a, b]\nf is differentiable on (a, b)\nSome c in (a, b) satisfies\nf′(c) = (f(b) − f(a)) / (b − a)\nONE TANGENT MATCHES THE SECANT\nHOW THE CHECKED ROUTE MOVES\n1 · Convert differentiability to pointwise derivatives\n2 · Apply Cauchy’s mean value theorem with the identity function\n3 · Rearrange at an interior witness c\nEXACT SCOPE\nReal-valued one-variable equality. No vector-valued equality and no endpoint witness." }, "visual": { "alt": "A smooth green curve crosses two gold endpoint values while a short green tangent through one interior witness is parallel to the gold endpoint secant.", "byteSize": 1932883, "caption": "An interior tangent matches the secant between the two endpoint values.", "description": "A low-text engraved curve schematic isolates the theorem's two endpoints, one strictly interior witness, and the matching directions of the endpoint secant and witness tangent.", "derivatives": [ { "byteSize": 17176, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-schematic-v2-640.webp", "sha256": "sha256:d305faf5b7f8831610e2cb9801fb7f596c3216b294305ae7185c82e85b385d2d", "width": 640 }, { "byteSize": 52728, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-schematic-v2-1200.webp", "sha256": "sha256:2f6b68974085bdc57791ce0014a3527bd7e91859c3ba4e0fb05931b530b482d3", "width": 1200 } ], "generatedAt": "2026-07-22T05:30:00Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/mean-value-theorem/theorem-schematic-v2.png", "sha256": "sha256:81c46f2698ba681e47ac23299fc2fcc8189881fb9e0838ef304a17601790a67d", "title": "Mean Value Theorem schematic", "width": 1536 }, "walkthrough": { "introduction": "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.", "pages": [ { "alt": "A green function ribbon and cobalt identity strand weave into an auxiliary ribbon whose two gold endpoint sockets sit at equal height.", "byteSize": 2838844, "caption": "A weighted difference of the function and the identity function has equal values at the two endpoints, preparing a Rolle witness.", "description": "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.", "derivatives": [ { "byteSize": 25280, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-01-v1-640.webp", "sha256": "sha256:de37a56e18e3d9238a45702c9efb69752c31afc7319b097dfe8ffaafce6b7f40", "width": 640 }, { "byteSize": 104322, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-01-v1-1200.webp", "sha256": "sha256:da75d56b6fdb9b9fd1333751f95245f8636a92142adfdd86fac3b5532c5a62de", "width": 1200 } ], "generatedAt": "2026-07-21T23:16:50Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-01-v1.png", "sha256": "sha256:fafa726e3fc2eb606de28fefe039bf407672d2477b9523b762ebc21bcdea4a44", "title": "Cauchy's auxiliary folds the endpoint differences together", "width": 1536, "equation": "h x = (g b - g a) * f x - (f b - f a) * g x and h a = h b", "kind": "proof_walkthrough_page", "sequenceIndex": 1 }, { "alt": "The woven equal-endpoint auxiliary carries one emerald interior witness lens with a short horizontal gold tangent beneath the recurring green and cobalt strands.", "byteSize": 2575235, "caption": "Continuity, interior differentiability, and equal endpoint values produce an interior point where the auxiliary derivative vanishes.", "description": "The source differentiates the auxiliary, proves it continuous on the closed interval, and invokes exists_hasDerivAt_eq_zero. Specializing the second function to the identity turns the resulting equality into the cross-multiplied Lagrange relation at one point strictly inside the interval.", "derivatives": [ { "byteSize": 23154, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-02-v1-640.webp", "sha256": "sha256:3fe43d85785009310c6c035c17f445f80d230c6fe4f2fc87089366a6fc710b12", "width": 640 }, { "byteSize": 90458, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-02-v1-1200.webp", "sha256": "sha256:8ce786d34391498516885bd40b30ec780eda16e13b1eb09823bccf63d72ce0a9", "width": 1200 } ], "generatedAt": "2026-07-22T01:04:16Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/mean-value-theorem/proof-walkthrough-02-v1.png", "sha256": "sha256:2ae2fd3f6e78e3d8717e216f6fe4cacf2b3fd652281d4513edcda425a42cb264", "title": "Rolle's theorem finds a horizontal auxiliary tangent", "width": 1536, "equation": "(b - a) * f' c = (f b - f a) * 1 for some c in (a,b)", "kind": "proof_walkthrough_page", "sequenceIndex": 2 }, { "alt": "A nonlinear green curve crosses an interior emerald witness where its green tangent is exactly parallel to the gold chord joining the endpoint values.", "byteSize": 2706425, "caption": "Division by the nonzero interval length and the derivative bridge turn the cross-multiplied witness into the exact selected equality.", "description": "Since a