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Rolle’s Theorem and the Mean Value Theorem. Guaranteeing Extrema. Rolle’s Theorem. Guarantees the existence of an extreme value in the interior of a closed interval. Rolle’s Theorem.

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Rolle’s Theorem and the Mean Value Theorem

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Rolle’s Theorem and the Mean Value Theorem

Guaranteeing Extrema


Rolle’s Theorem

  • Guarantees the existence of an extreme value in the interior of a closed interval


Rolle’s Theorem

  • Let f be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). If

  • f(a) = f(b)

  • then there is at least one number c in (a, b) such that f’(c) = 0.


Rolle’s Theorem

  • Three things that must be true for the theorem to hold:

  • (a) the function must be continuous

  • (b) the function must be differentiable

  • (c) f(a) must equal f(b)


Rolle’s Theorem Examples

  • Book

  • On-line


Mean Value Theorem

  • If f is continuous on the closed interval

  • [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that


Mean Value Theorem

  • In other words, the derivative equals the slope of the line.


Mean Value Theorem

  • Tutorial

  • Examples

  • More Examples


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