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CHAPTER 3 Applications of Differentiation 220 23. (a) Show that a polynomial of degree 3 has at most three real roots. 4. Draw the graph of a function that is continuous on [0, 8] (b) Show that a polynomial of degree n has at most n real where f(0) = 1 and f(8) = 4 and that does not satisfy the roots. conclusion of the Mean Value Theorem on [0, 8]. 5-8 Verify that the function satisfies the three hypotheses of 24. (a) Suppose that f is differentiable on R and has two roots. Show that f' has at least one root. Rolle's Theorem on the given interval. Then find all numbers c (b) Suppose f is twice differentiable on R and has three that satisfy the conclusion of Rolle's Theorem. roots. Show that f" has at least one real root. 5. f (x) = 2x2 - 4x + 5, [-1, 3] (c) Can you generalize parts (a) and (b)? 6. f(x) = x3 - 2x2 - 4x+ 2, [-2, 2] 25. If f(1) = 10 and f'(x) > 2 for 1 0 17. Let f(x) = (x - 3)-2. Show that there is no value of c in (1, 4) such that f(4) - f(1) = f'(c)(4 - 1). Why does this 1 + if x