Standard Version of The Theorem
If a real-valued function ƒ is continuous on a closed interval, differentiable on the open interval (a, b), and ƒ(a) = ƒ(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 the Taylor's theorem.
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