1. Prove the second part of Theorem 12.2. That is, let I 1; 1R be an open interval and suppose f t I z~ R is a. function. Show that if f has a local maximum at a point c E I and f is differentiable at cJ then f'(c) = D. 2. (i) Use Precalculus methods to nd the two :r-intercepts of the function: n) =c245c+3 (ii) Use Rolle's Theorem to show that f'{:c) = [1 at some point :c = c between the two :s-intercepts. 3. Consider the function f: R i R, given by rs} = 1:4 2:112 + 1. Use Rolle's Theorem to show that there is at least one point :r. = c in the open interval (2. 2) such that f'(c) = 0. Find all such points c on the interval {2. 2]. Which ones are critical points? Your justication should be accompanied by a graph of f, showing intercepts and critical points. 4. (i) Show that the Mean 1tiralue Theorem implies Rolle's Theorem. (ii) Use the Mean Value Theorem to prove that cI 3r 1 + :E for all :r '.:- i] (assume that the derivative of e3 is known to be ex). Hint: Consider the fnnction re) = e'\" .r for oil :1: :5 i]