Rolle's Theorem

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.
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Standard version of the theorem [edit]
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.
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History [edit]
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]
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Examples [edit]
First example [edit]

A semicircle of radius r.
For a radius r > 0, consider the function

Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval [−r,r] and differentiable in the open interval (−r,r), but not differentiable at the endpoints −r and r. Since f(−r) = f(r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. Note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.

Second example [edit]

The graph of the absolute value function.
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold.