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# Rolle’s Theorem and the Mean Value Theorem - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Rolle’s Theorem and the Mean Value Theorem' - tobias-warren

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### Rolle’s Theorem and the Mean Value Theorem

Guaranteeing Extrema

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

• 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.

• 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)

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• On-line

• 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

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

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