Local Maximum Property Problem: The definition of local maximum is as follows: f(c) is a

Chapter 8, Problem 31

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Local Maximum Property Problem: The definition of local maximum is as follows: f(c) is a local maximum of f on the interval (a, b) if and only if f(c) f(x) for all values of x in (a, b). Figure 8-3cc illustrates this definition. Figure 8-3cc a. Prove that if f(c) is a local maximum of f on (a, b) and f is differentiable at x = c in (a, b), then c) = 0. (Consider the sign of the difference quotient when x is to the left and to the right of c, then take left and right limits.) b. Explain why the property in part a would be false without the hypothesis that f is differentiable at x = c. c. Explain why the converse of the property in part a is false.