Cauchy’s mean value theorem

E825423

mean value theorem theorem in real analysis

Cauchy’s mean value theorem is a fundamental result in real analysis that generalizes the standard mean value theorem by relating the rates of change of two differentiable functions on an interval.

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All labels observed (3)

Label	Occurrences
Cauchy’s mean value theorem canonical	2
Cauchy mean value theorem	1
L’Hôpital’s rule	1

How this entity was disambiguated

This entity first appeared as the object of triple T9843487 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Cauchy’s mean value theorem
Context triple: [Augustin-Louis Cauchy, notableFor, Cauchy’s mean value theorem]

A. Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
B. Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.
C. Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Karamata's inequality
Karamata's inequality is a fundamental result in majorization theory that generalizes several classical inequalities by comparing sums of convex (or concave) functions over majorized sequences.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Cauchy’s mean value theorem
Target entity description: Cauchy’s mean value theorem is a fundamental result in real analysis that generalizes the standard mean value theorem by relating the rates of change of two differentiable functions on an interval.

A. Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
B. Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.
C. Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Karamata's inequality
Karamata's inequality is a fundamental result in majorization theory that generalizes several classical inequalities by comparing sums of convex (or concave) functions over majorized sequences.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mean value theorem ⓘ theorem in real analysis ⓘ
appearsIn	standard calculus textbooks ⓘ standard undergraduate analysis textbooks ⓘ
appliesTo	differentiable functions ⓘ real-valued functions ⓘ
assumes	g′(x) ≠ 0 for some x in (a,b) when used to form a ratio ⓘ two real-valued functions f and g ⓘ
assumptionOnInterval	a < b ⓘ
auxiliaryFunctionExample	h(x) = (f(b)−f(a))·g(x) − (g(b)−g(a))·f(x) ⓘ
category	theorems about continuous functions ⓘ theorems about derivatives ⓘ
conclusion	there exists c in (a,b) such that (f(b)−f(a))·g′(c) = (g(b)−g(a))·f′(c) ⓘ there exists c in (a,b) such that f′(c)/g′(c) = (f(b)−f(a))/(g(b)−g(a)) when g(b) ≠ g(a) and g′(c) ≠ 0 ⓘ
domainCondition	functions continuous on [a,b] ⓘ functions defined on a closed interval [a,b] ⓘ functions differentiable on (a,b) ⓘ
field	calculus ⓘ real analysis ⓘ
generalizes	Lagrange’s mean value theorem ⓘ standard mean value theorem ⓘ
hasSpecialCase	Lagrange’s mean value theorem ⓘ Rolle’s theorem ⓘ
historicalPeriod	19th-century mathematics ⓘ
holdsIn	real line ℝ ⓘ
implies	existence of a point where relative rates of change match endpoint increments ⓘ
logicalForm	existence theorem ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
proofMethod	application of Rolle’s theorem to an auxiliary function ⓘ
relatedTo	Rolle’s theorem ⓘ Taylor’s theorem ⓘ linked to: Taylor series l’Hôpital’s rule ⓘ linked to: L’Hôpital’s rule for indeterminate limits
requires	continuity of f and g on [a,b] ⓘ differentiability of f and g on (a,b) ⓘ real-valued functions on [a,b] ⓘ
specialCaseOf	mean value theorem for derivatives ⓘ
typicalNotation	∃c ∈ (a,b) such that (f(b)−f(a))·g′(c) = (g(b)−g(a))·f′(c) ⓘ
usedFor	comparing growth rates of functions ⓘ deriving error estimates in numerical analysis ⓘ establishing inequalities in analysis ⓘ proving l’Hôpital’s rule ⓘ proving properties of inverse trigonometric functions ⓘ proving properties of logarithmic and exponential functions ⓘ
usedIn	analysis of monotonicity and convexity via ratios of derivatives ⓘ elementary proofs of l’Hôpital’s rule ⓘ real-variable theory ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy → notableFor → Cauchy’s mean value theorem ⓘ

subject linked to: Augustin-Louis

Guillaume de l’Hôpital → knownFor → L’Hôpital’s rule ⓘ

linked to: Cauchy’s mean value theorem

L’Hôpital’s rule for indeterminate limits → generalizationOf → Cauchy’s mean value theorem ⓘ

L’Hôpital’s rule for indeterminate limits → proofUses → Cauchy mean value theorem ⓘ