Rolle's Theorem

In calculus, it is the branch of mathematics; Rolle's Theorem essentially states that a differentiable function, which obtains equal values at 2 points, should have a point somewhere between them where the slope of the tangent line to the graph of function is zero.

Then there is at least 1 point c ε (a,b) such that f'© = 0

If a and b are 2 roots of a polynomial equation f(x) = 0 then Rolle's Theorem says that there is at least one root c between a and b for f'(x) = 0.

The converse of the theorem is not true.

If a real-valued function ƒ is continuous on a closed interval [a, b], is differentiable on open interval (a, b), and ƒ(a) = ƒ(b), then there is a c in the open interval (a, b) such that  f ' (c) = 0

This version of Rolle's Theorem is used to prove the mean value theorem, of which Rolle's Theorem is indeed a special case.

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