25.4 Problem. In calculus, we usually apply derivative tests for extreme values occurring at interior points of intervals, so here is a chance to think about what happens at the endpoints. Let f = C² ([a, b]) and suppose that f(a) = max f(x). a≤x≤b (i) Use the definition of the derivative to prove that f'(a) ≥ 0. (ii) Give an example to show that we may have f'(a) > 0, in contrast to our likely calculus intuition that f'(a) = 0. (iii) Show that ƒ"(a) ≤ 0 by contradiction as follows. If ƒ"(a) > 0, then by continuity f"(x) > 0 for a ≤ x ≤ a + d, with > 0 sufficiently small. Use FTC2 to show that f'(x) > 0 for a≤x≤a+d and thus f is strictly increasing on [a, a + 6]. How does this contradict the maximum occurring at a? (iv) Give examples to show that both f" (a) < 0 and f"(a) = 0 are possible.