Cauchy’s mean value theorem
E825423
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy’s mean value theorem canonical | 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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy’s mean value theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy
this entity surface form:
L’Hôpital’s rule