CCe M O Bb Upload Assignment: Ch. 3.8 - CA X + O File C:/Users/Aisha/Downloads/6083.%20Calculus_%2011th%20Edition_%20Howard%20Anton%20_%20lrl%20C.%20Bivens%20_%20Stephen%20Davis.pdf for all values of x and y in the interval. (b) Use the result in part (a) to show that |sinx – sin y < |x – y| priate picture. 36. (a) Prove that if f"(x) > 0 for all x in (a, b), then f'(x) = 0 at most once in (a, b). (b) Give a geometric interpretation of the result in (a). for all real values of x and y. 26. (a) Use the Mean-Value Theorem to show that if f is differentiable on an open interval, and if |f (x)| > M for all values of x in the interval, then |f(x) – fV)| > M|x – y| for all values of x and y in the interval. 37. (a) Prove part (b) of Theorem 3.1.2. (b) Prove part (c) of Theorem 3.1.2. 38. Use the Mean-Value Theorem to prove the following result: Let f be continuous at xo and suppose that lim, exists. Then f is differentiable at xo, of'(x) (b) Use the result in part (a) to show that and |tan x – tan y| > |x – y| for all values of x and y in the interval (–x/2, 7/2). f'(xo) = lim f'(x) (c) Use the result in part (b) to show that [Hint: The derivative f'(xo) is given by |tan x+ tan y| > |x+ y| for all values of x and y in the interval (-7/2, 7/2). f(x) – f(xo) f'(xo) = lim provided this limit exists.] 27. (a) Use the Mean-Value Theorem to show that FOCUS ON CONCEPTS Vỹ - Vĩ < 2Vx 39. Let if 0 < x < y. f(x) = {3x?, lax + b, x > 1 (b) Use the result in part (a) to show that if 0 < x < y, then Vay < (x+y). Find the values of a and b so that f will be differentiable at x = 1. 28. Show that if f is differentiable on an open interval and f'(x) 0 on the interval, the equation f(x) = 0 can have 40. (a) Let at most one real root in the interval. f(x) = %3D x + 1, 29. Use the result in Exercise 28 to show the following: (a) The equation x'+4x - 1 = 0 has exactly one real Show that lim f'(x) = lim f'(x) root. (b) If b2 - 3ac < 0 and if a 0, then the equation ax + bx+ cx + d 0 but that f' (0) does not exist. (b) Let has exactly one real root. Jx², x<0 x', x> 0 f(x) = 30. Use the inequality V3 < 1.8 to prove that 1.7 < V3 < 1.75 Show that f'(0) exists but f"(0) does not. [Hint: Let f(x) = Vx, a = 3, and b = 4 in the Mean-Value Theorem.] %3D 41. Use the Mean-Value Theorem to prove the following result: The graph of a function f has a point of vertical tangency at (xo, f(xo)) if f is continuous at xo and f'(x) approaches either +o or -o as xxo and as xXo . 31. Use the Mean-Value Theorem to prove that < sin x 0) 32. (a) Show that if f and g are functions for which f'(x) = g(x) and g'(x), = f(x) for all x, then f²(x) - g²(x) is a constant. 42. Writing Suppose that p(x) is a nonconstant polynomial with zeros at x = a and x = b. Explain how both the Extreme-Value Theorem (3.4.2) and Rolle's Theorem can 33. (a) Show that if f and g are functions for which f'(x) = g(x) and g'(x) = -f(x) for all x, then f²(x) + g²(x) is a constant. (b) Give an example of functions f and g with this property. be used to show that p has a critical point between a and b. 43. Writing Find and describe a physical situation that illus- trates the Mean-Value Theorem. V QUICK CHECK ANSWERS 3.8 1. (a) [0, 1] (b) c = } 2. [-3,3]; c = -2,0, 2 3. (a) b= 2 (b) c = 1