Suppose that f : [a, b] to R is differentiable and c is an element of [a, b]. Th

Question

Suppose that f : [a, b] to R is differentiable and c is an element of [a, b]. Then show that there exists a sequence {x_n} converges to c where x_n does not equal c for all n, such that f '(c) = lim f ' (x_n) Note: I use the definition of differentiation of: lim (x goes to c) (f(x) - f(c)) / (x - c). Second Note: I have previously asked the question and it seemed to be answered, but it wasn't visible. I'm not sure why. Time ran out though. So, if you answer it, please make sure it can be seen. Either that, or the answer given had nothing to do with the question.

Explanation / Answer

Let h(x) = f(x) - g(x). Note that h(a) = f(a) - g(a) = 0. It is easy to check that h is continuous on [a,b] and differentiable on (a,b). Thus the Mean Value Theorem applies, and we get [h(b) - h(a)] / (b - a) = h'(c) for some c in (a,b). ==> [(f(b) - g(b)) - 0] / (b - a) = f'(c) - g'(c) < 0, since f '(x)

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