Objectives: Be able to determine if Rolle’s Theorem can be applied to a

1. Rolle's Theorem • Objectives: • Be able to determine if Rolle’s Theorem can be applied to a • function on a closed interval. • 2. Be able to apply Rolle’s Theorem to various functions. Critical Vocabulary: Rolle’s Theorem

2. If f is continuous on a closed interval [a, b], then f has both a minimum and maximum on the interval. However, the minimum and/or maximum may occur at the endpoints Example 1: Absolute Max: _______ Absolute Min: ________

3. Rolle’s Theorem (named after French mathematician Michel Rolle) gives us conditions that guarantees the existence of an extreme value on the interior of the closed interval. Let f be continuous on a 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 (horizontal tangent line). Example 2: Determine if Rolle’s Theorem can be applied to the function f(x) = x2 + 6x + 13; [-7, 1]

4. Let f be continuous on a 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 (horizontal tangent line). Example 3: Determine if Rolle’s Theorem can be applied to the function

5. Example 4: Find the two x-intercepts of the function f and show that f’(x) = 0 at some point between the two intercepts.

6. Example 5 (Problem 10 on page 326): Determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all the values of c in the open interval (a,b) such that f’(c) = 0. f(x) = (x - 3)(x + 1)2; [-1, 3]

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