# Rolle's Theorem

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• Topic: Calculus, Derivative, Continuous function
• Pages : 3 (568 words )
• Published : May 13, 2013

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n calculus, Rolle's theorem essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. -------------------------------------------------

Standard version of the theorem 
If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a c in the open interval (a, b) such that

This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem. -------------------------------------------------

History 
Indian mathematician Bhāskara II (1114–1185) is credited with knowledge of Rolle's theorem.[1] The first known formal proof was offered by Michel Rolle in 1691, which used the methods of differential calculus. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.[2] -------------------------------------------------

Examples 
First example