Proof of the Local Extreme Value Theorem Prove Theorem 4.2

Chapter 4, Problem 83

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Proof of the Local Extreme Value Theorem Prove Theorem 4.2 for a local maximum: If f has a local maximum value at the point c and f _1c2 exists, then f _1c2 = 0. Use the following steps. a. Suppose f has a local maximum at c. What is the sign of f 1x2 - f 1c2 if x is near c and x 7 c? What is the sign of f 1x2 - f 1c2 if x is near c and x 6 c? b. If f _1c2 exists, then it is defined by lim xSc f 1x2 - f 1c2 x - c . Examine this limit as xS c + and conclude that f _1c2 0. c. Examine the limit in part (b) as xS c- and conclude that f _1c2 0. d. Combine parts (b) and (c) to conclude that f _1c2 = 0.