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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Observed surface forms (1)

Surface form	Occurrences
L’Hôpital’s rule	1

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 NERFINISHED ⓘ Rolle’s theorem NERFINISHED ⓘ
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 NERFINISHED ⓘ
proofMethod	application of Rolle’s theorem to an auxiliary function ⓘ
relatedTo	Rolle’s theorem NERFINISHED ⓘ Taylor’s theorem NERFINISHED ⓘ l’Hôpital’s rule NERFINISHED ⓘ
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 ⓘ

Referenced by (2)

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

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

subject surface form: Augustin-Louis Cauchy

Guillaume de l’Hôpital → knownFor → Cauchy’s mean value theorem ⓘ

this entity surface form: L’Hôpital’s rule