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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.

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)

• 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