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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

- 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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- 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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