3. a) A function is said to be (weakly) increasing on an interval [a, b] if for all c < d in [a, b], we have f(c) = f(d)

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3 A A Function Is Said To Be Weakly Increasing On An Interval A B If For All C D In A B We Have F C F D 1 (219.33 KiB) Viewed 41 times

3. a) A function is said to be (weakly) increasing on an interval [a, b] if for all c < d in [a, b], we have f(c) = f(d). Suppose f is differentiable on [a, b]. Show that f is weakly increasing on [a, b] if and only if f'(x) > 0 for all x € [a, b]. Hint: Prove the forward direction directly. For the backward direction, prove the contrapositive using the mean value theorem. b) Suppose that f and g are differentiable functions on R. Suppose that f(0) = g(0) and that f'(x) < g'(x) for all x > 0. Show that f(x) = g(x) for all x > 0. Hint: Apply part (a) of the previous problem to a function involving both f and g.