SECTION 4.2 THE MEAN VALUE THEOREM 239 (b) Suppose f is twice differentiable on [FE and has three roots. Show that f" has at least one real root. (c) Can you generalize parts (a) and (b)? e I/ff(l)=10 and f'(x) .3: 2 for l E .i: a 4, how small can "f(4) possibly be? 24. Suppose that 3 5-5 f '(x) a: 5 for all values of x. Show that 18 e e) f(2) s; 30. I 25?}Poes there exist a function f such that f(0) = l, f (2) = 4, n/ and f'(x) E 2 for all x? 25. Suppose that f and g are continuous on [(1, b] and differ- entiable on (a, b). Suppose also that n) = 9(a) and f'(x) 0. _ 28. Suppose f is an odd function and is differentiable every- where. Prove that for every positive number b, there exists a number 0 in (-b, b) such that f'(c) = f (b)/ b