4.2 The Mean Value Theorem 235 Proof of Law 1 Let y1 = er and (4) Then x] = In y, and x2 = In yz Take logs of both sides of Eqr. (4). *+ x = Iny + In yz = In y1 )'2 Product Rule for logarithms exitx = elnyi )2 Exponentiale. = y1 Y2 gloa = u = exless. The proof of Law 4 is similar. Laws 2 and 3 follow from Law 1 (Exercises 77 and 78). EXERCISES 4.2 Checking the Mean Value Theorem is zero at x = 0 and x = 1 and differentiable on (0, 1), but its de- Find the value or values of c that satisfy the equation rivative on (0, 1) is never zero. How can this be? Doesn't Rolle's Theorem say the derivative has to be zero somewhere in (0, 1)? f (b) - f(a) 2 = f'(c) Give reasons for your answer. b - a 16. For what values of a, n, and b does the function in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1-8. x = 0 1. f(x) = x2+ 2x - 1, [0, 1] f ( x ) = 3 - x2 + 3x + a . 0 0 throughout an interval [a, b ], then f' has 13. f (x) = x - x, - 25 x5- 1 at most one zero in [a, b ]. What if f"