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)

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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.

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