If a real function is continuous on a closed interval, differentiable on its interior, and takes equal values at the two endpoints, then its derivative vanishes at some point in between. Named for Michel Rolle, it is the special case from which the mean value theorem is derived.
Facts
StatementRolle's theorem states that a real-valued differentiable function that takes equal values at two distinct points must have a stationary point somewhere between them, a point where its derivative is zero. 1 Classification
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Source Michel Rolle (Wikipedia)
Proved By
Source Rolle's Theorem (Wikipedia)
Sources
1. Rolle's Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary point somewhere between them, that is, a point where its derivative is zero.
History section, on Michel Rolle's own proof
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions.
Proved By: Augustin-Louis Cauchy, Lead
The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem.
View the Source Michel Rolle (Wikipedia)
Principal Result Of: Michel Rolle, Lead, principal resultQuote, Principal Result Of: Michel Rolle, Lead, principal result
He is best known for Rolle's theorem (1691).
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