Exercise 8:[Hard] Let f be a function differentiable on (c-p, c+p) for some p > 0. Suppose f has a maximum at c, ie., f(c)2 f(x) for all x € (c - p,c+ p) Last modified: October 29, 2019, Due: November 6, 2019. 1 ΜΑΤΗ 15100, SECTION 13 2 (a) Prove that f'(c) < 0. f(c+h)-f(c) Hint: Consider the limit: lim0+ What can be said about the numerator? (b) Prove that f'(c) > 0. f(c+h)-f(c) Hint: Consider the limit: lim/0- (c) Conclude that f'(c) = 0. Remark: The same result is true for minima as well, with basically the same proof. Note this is not an if and only if statement: let g(x) = x3. Then g'(0) = 0, but g has neither a maximum nor minimum at x = 0

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