SECTION 2.3 Properties of Functions 97 In Problems 49–56, for each graph of a function y f x , ( ) = find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 49. (5, 1) (3, 3) (2, 2) (1, 4) 1 3 5 x y 4 2 50. (5, 0) (4, 4) (1, 1) (0, 2) 1 3 5 x y 4 2 51. (4, 3) (3, 4) (1, 1) (0, 3) 1 3 5 x y 4 2 52. (0, 1) (1, 3) (2, 4) (3, 2) 1 3 x y 4 2 53. (0, 0) (2, 3) (3, 2) 1 3 x y 4 2 54. (2, 4) (0, 2) 1 3 x y 4 2 55. (2, 0) (3, 2) (4, 1) 1 3 x y 2 56. (0, 2) (1, 3) (2, 0) (3, 1) 1 3 x y 2 In Problems 57–64, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 57. f x x x3 2 2, 2 3 ( ) [ ] = − + − 58. f x x x3 5 1, 3 3 2 ( ) [ ] = − + − 59. f x x x 2, 2 5 3 ( ) [ ] = − − 60. f x x x 2, 2 4 2 ( ) [ ] = − − 61. f x x x x 0.2 0.6 4 6 6, 4 3 2 ( ) [ ] = − − + − − 62. f x x x x 0.4 0.6 3 2 4, 5 3 2 ( ) [ ] = − + + − − 63. f x x x x 0.25 0.3 0.9 3 3, 2 4 3 2 ( ) [ ] = + − + − 64. f x x x x 0.4 0.5 0.8 2 3, 2 4 3 2 ( ) [ ] = − − + − − 65. Find the average rate of change of f x x2 4: 2 ( ) = − + (a) From 0 to 2 (b) From 1 to 3 (c) From 1 to 4 66. Find the average rate of change of f x x 1: 3 ( ) = − + (a) From 0 to 2 (b) From 1 to 3 (c) From 1− to 1 67. Find the average rate of change of g x x x4 7: 3 ( ) = − + (a) From 3− to 2− (b) From 1− to 1 (c) From 1 to 3 68. Find the average rate of change of h x x x2 3: 2 ( ) = − + (a) From 1− to 1 (b) From 0 to 2 (c) From 2 to 5 69. f x x5 2 ( ) = − (a) Find the average rate of change from 1 to 3. (b) Find an equation of the secant line containing f 1, 1 ( ) ( ) and f 3, 3. ( ) ( ) 70. f x x4 1 ( ) = − + (a) Find the average rate of change from 2 to 5. (b) Find an equation of the secant line containing f 2, 2 ( ) ( ) and f 5, 5. ( ) ( ) 71. g x x 2 2 ( ) = − (a) Find the average rate of change from 2− to 1. (b) Find an equation of the secant line containing g 2, 2 ( ) ( ) − − and g 1, 1. ( ) ( ) 72. g x x 1 2 ( ) = + (a) Find the average rate of change from 1− to 2. (b) Find an equation of the secant line containing g 1, 1 ( ) ( ) − − and g 2, 2. ( ) ( ) 73. h x x x2 2 ( ) = − (a) Find the average rate of change from 2 to 4. (b) Find an equation of the secant line containing h 2, 2 ( ( )) and h 4, 4. ( ( )) 74. h x x x 2 2 ( ) = − + (a) Find the average rate of change from 0 to 3. (b) Find an equation of the secant line containing h 0, 0 ( ( )) and h 3, 3. ( ( )) 75. Mixed Practice g x x x 27 3 ( ) = − (a) Determine whether g is even, odd, or neither. (b) There is a local minimum value of 54 − at 3. Determine the local maximum value. 76. Mixed Practice f x x x 12 3 ( ) = − + (a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2. Determine the local minimum value. 37. f x x4 3 ( ) = 38. f x x x 2 4 2 ( ) = − 39. g x x 10 2 ( ) = − 40. h x x3 5 3 ( ) = + 41. F x x4 3 ( ) = 42. G x x ( ) = 43. f x x x ( ) = + 44. f x x2 1 3 2 ( ) = + 45. g x x 1 8 2 ( ) = + 46. h x x x 1 2 ( ) = − 47. h x x x3 9 3 2 ( ) = − − 48. F x x x 2 ( ) = In Problems 37–48, determine algebraically whether each function is even, odd, or neither.
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