Rolle's theorem

In calculus, Rolle's theorem states that if a function f is continuous on a closed interval and differentiable on the open interval (a,b), and f(a) = f(b) then there is some number c in the open interval (a,b) such that

Related Topics:
Calculus - Continuous - Differentiable

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:f '(c) = 0.

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Intuitively, this means that if a smooth curve is equal at two points then there must be a stationary point somewhere between them. All the assumptions are necessary. For example, if f(x) = |x|, the absolute value of x, then we have that f(-1) = f(1), but there is no x between -1 and 1 for which f '(x) = 0. This is because that function, although continuous, is not differentiable.

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The theorem was first stated by Michel Rolle, and published in 1691.

Related Topics:
Michel Rolle - 1691

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Rolle's Theorem is used in proving the mean value theorem, which eliminates the requirement that f(a) = f(b).

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