248 CHAPTER 4 . APPLICATIONS OF THE DERIVATIVE 58. f ( x ) = ex - 2x on [0 21. f'(1) and f'(3) are undefined; f' (2) = 0; f has a local maxi- 4x3 mum at x = 1; f has a local minimum at x = 2; f has an absolute "+ 5x2 3 maximum at x = 3; and f has an absolute minimum at x = 4. 59 . f (x ) = 22. f' (x) = 0 at x = 1 and 3; f' (2) is undefined; f has an absolute 60. f(x) = 2x6 - 15x4 maximum at x = 2; f has neither a local maximum nor a local X minimum at x = 1; and f has an absolute minimum at x = 3. 61. f(x) = - ( x 2 + 9 ) 5 on Practice Exercises 23-42. Locating critical points Find the critical points of the follow- 62. f(x ) = *1/2( * - 4 ing functions. Assume a is a nonzero constant. 63. f(x) = sec x on 23. f ( x ) = 3x2 - 4x+ 2 24. f(x) =-x'. 25 . f ( x ) = 1 - 9x x3 64. f ( x) = x1/3(x + 4) 0 26. f(x ) = - 3x2 + 10 4 3 65. f(x) = x3e * on [-1, 27. f ( x ) = 3x3 + 3x2 28. f(x ) = 4x5 2 - 2x 5 - 3x3 + 5 67. f(x ) = x2/3 (4 - x2) 29. f(x ) = x3 - 402 x 30. ) f ( x ) = x - 5 tan-1x 3t 68. f (t) =- - on [-2 31. f (t) = 7 12 + 1 1 2 + 1 32. f ( x ) = 12x5 - 20*3 69. Efficiency of wind tur 33 . f ( x ) = - etex 2 34. f(x) = sin x cos x into electrical power. wind before it encount 35. f ( x ) = -+Inx 36. f (t ) = 12 - 2 in(12 + 1) downstream velocity swept out by the turkin 37. f (x ) = x2 V x+ 5 38, f(x) = (sin x ) (cos x) 39 . f ( x ) = x V x - a 40. 1 (x ) = - Vx - 1 75 - att 41. f ( t ) = 15 42. f (x ) = x3 - 3ax2 + 302x - a3 43-68. Absolute maxima and minima Determine the location and value of the absolute extreme values of f on the given interval, if they exist. 43. f (x ) = x2 - 10 on [-2, 3 ] 44.) f ( x ) = ( x + 1 ) 4/3 on [ - 9, 7] 45. f ( x) = x3 - 3x2 on [ - 1, 3] 46. f (x ) = x4 - 4x3 + 4x2 on [-1, 3] a. Assuming that v > 47. f( x ) = 3x5 - 25x3 + 60x on [-2, 3] that 0 5 2 . V 1 S 1 . 48. f ( x ) = 2ex - x2 on [0, 2e ] b. The amount of pow 49. f (x) = cos2 x on [ 0, TT ] ratio r = -, the ra 50 . f ( x ) = - V1 ( x 2 + 3) 2 on [ - 2 , 2] velocity. Let R(r) e from the total availa 51. f(x) = sin 3x on [-17 / 4, TT / 3 ] value of r. In about 52. f ( x ) = 3x2/3