Prove Theorem 4.2 for a local maximum: If f has a local maximum value at the point
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Prove Theorem 4.2 for a local maximum: If f has a local maximum value at the point c and f'(c) exists, then f'(c) = 0. Use the following steps.
a. Suppose f has a local maximum at c. What is the sign of f(x) - f(c) if x is near c and x > c? What is the sign of f(x) - f(c) if x is near c and x < c?
b. If f'(c) exists, then it is defined byExamine this limit as x → c+ and conclude that f'(c) ≤ 0.
c. Examine the limit in part (b) as x → c- and conclude that f'(c) ≥ 0.
d. Combine parts (b) and (c) to conclude that f'(c) = 0.
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Related Book For
Calculus Early Transcendentals
ISBN: 978-0321947345
2nd edition
Authors: William L. Briggs, Lyle Cochran, Bernard Gillett
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