228 Chapter 4 Applications of Derivatives Absolute Extrema on Finite Closed Intervals In Exercises 21-40, find the absolute maximum and minimum values 57. y = x2/3 (x+ 2) 58. y = x2/3(x2 - 4) of each function on the given interval. Then graph the function, Iden- 59. y = xV4 - x 60. y = x2V3-x tify the points on the graph where the absolute extrema occur, and include their coordinates. 61. y = 14 - 27, x 5 1 1x+ 1, x > 1 21. f (x) = 2x - 5, -25x53 62. y = 12 13 - x, * 1 25. F(x) = - - 2 0.5 5 x 5 2 64. ). = 3 -4x2- 2x + 15 . x51 26. F ( x ) = - 4, -25x5-1 3 - 617 + 8x , * > 1 27 . h( x ) = Vx - 1= x=8 In Exercises 65 and 66, give reasons for your answers. 28 . h(x ) = -3x2/3, -15 x= 1 65. Let f(x) = (x - 2)213 29. 8 (x) = V4 - x, -25x51 a. Does f'(2) exist? 30. 8(x) = -V5- x, -VS=x50 b. Show that the only local extreme value of f occurs at x = 2 31. f(0) = sin 0, c. Does the result in part (b) contradict the Extreme Value 6 Theorem? 32. f(0) = tan 0, d. Repeat parts (a) and (b) for f(x) = (x - a) /5, replacing 2 by a. 66. Let f(x) = 1x3 - 9x1. 33. 8(x) = CSC X, 3 a. Does f'(0) exist? b. Does f'(3) exist? 34. 8 (x ) = sec x, - 2 5xs" c. Does f'(-3) exist? d. Determine all extrema of f. 35. f(1) = 2 - 181, -15153 In Exercises 67-70, show that the function has neither an absolute 36. f (1) = 12 - 51, 45157 minimum nor an absolute maximum on its natural domain. 37. 8(x) = xe z, - 1 5 x 5 1 67. y = xl' + x+ x-5 68. y = 3x + tan x 38. h(x) = In(x + 1), 05x53 69. y = 1- e 39. f ( x ) = - + Inx, 0.5 5 x 5 4 ex+ 1 70. y = 2x - sin 2x 40. 8 ( x ) = er, -25x=1 Theory and Examples In Exercises 41-44, find the function's absolute maximum and mini- 71. A minimum with no derivative The function f(x) = [x] has an mum values and say where they occur. absolute minimum value at x = 0 even though f is not differen- 41. f(x ) = x4/3, -1 5x= 8 tiable at x = 0. Is this consistent with Theorem 2? Give reasons for your answer. 42. f( x ) = x313, - 15 x 5 8 72. Even functions If an even function f(x) has a local maximum 43. 8(0) = 03/5, -32 0 = 1 value at x = c, can anything be said about the value of f at 44. h(0) = 30213, - 27 50 58 x = -c? Give reasons for your answer. Finding Critical Points 73. Odd functions If an odd function g(x) has a local minimum val- In Exercises 45-56, determine all critical points for each function. ue at x = c, can anything be said about the value of g at x = -c? Give reasons for your answer. 45. y = x2 - 6x + 7 46. f(x) = 6x2 - 13 74. No critical points or endpoints exist We know how to find the 47. f ( x ) = x(4 - x) 3 48. g(x) = (x - 1)3(x- 3)2 extreme values of a continuous function f(x) by investigating its values at critical points and endpoints. But what if there are no 49. y = x2+ 50. f(x) = critical points or endpoints? What happens then? Do such func- 51. y = 12 - 32VX 52. 8 (x) = V2x - x2 tions really exist? Give reasons for your answers. 53. y = In (x + 1) - tan lx 54. y = 2V/1 - x2 + sin ] x 75. The function 55. y = x3 + 3x2 - 24x + 7 56. y = x-3x2/3 V (x ) = x(10 - 2x) (16 - 2x). 0