- 12 A Sketch The Graph Of A Function On 1 2 That Has An Absolute Maximum But No Local Maximum B Sketch The Grap 1 (185.5 KiB) Viewed 48 times
12. (a) Sketch the graph of a function on [-1, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [-1, 2] that has a local maximum but no absolute maximum. 13. (a) Sketch the graph of a function on [-1, 2] that has an absolute maximum but no absolute minimum. Sketch the graph of a function on [-1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum. (b) (b) 14. (a) Sketch the graph of a function that has two local max- ima, one local minimum, and no absolute minimum. Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers. 15-28 Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) 15. f(x)=3-2x, x>-1 16. f(x)=x², -1<x<2 17. f(x)=1/x, x>1 18. f(x)=1/x, 1<x<3 19. f(x) = sinx, 0<x< π/2 20. f(x) = sinx, 0<x< π/2 21. f(x) = sinx, 22. f(t)= cos t, 23. f(x) = ln x, 25. f(x) = 1 π/2 ≤ x ≤ π/2 -3π/2 ≤ t ≤ 3TT/2 0 < x < 2 √x -{2² if -1 ≤x≤ 0 if 0<x< 1 28. f(x) = {2-21 #f1<x<3 2x + if 1 27. f(x) = 2-3x 37. p(x) = 29-48 Find the critical numbers of the function. 29. f(x) 3x² + x2 31. f(x) = 3x² + 8x³48x² 33. g(t)t + 51³ + 50t 35. g(y) y-1 y²-y + 1 24. f(x) = |x| 26. f(x) = ex x² + 2 2x 1 39. h(t)t3/4-211/4 41. F(x)= x/5(x-4)² 43. f(x)= x¹(4- x)²/3 30. g(v) v³ 12v + 4 32. f(x) = 2x³ + x² + 8x 34. A(x)=32x| 36. h(p) = P-1 p² +4 t² +9 1². 9 38. q(t)= 40. g(x)=√√4x² 42. h(x) = x ¹/³(x - 2) 44. f(0) = 0 + √2 cos 0 45. f(0) 2 cos 0 sin²0 47. g(x)=x² ln x 49-50 A formula for the derivative of a function f is given. How many critical numbers does f have? 49. f'(x) 5e-0.1|x sinx 1 50. f'(x) = 46. p(t) te 48. B(u) 4 tan u u 57. f(x) = x + x + — [0.2, 4] 51-66 Find the absolute maximum and absolute minimum values off on the given interval. 51. f(x) = 12 + 4x x², [0, 51 52. f(x) = 5 + 54x2x³, [0, 4] 53. f(x) = 2x³ 3x² 12x + 1, [-2,3] 54. f(x)= x³6x² + 5, [-3, 51 55. f(x) = 3x4 - 4x³ − 12x² + 1, [-2,3] 56. f(t)= (t²-4)³, [-2,3] 58. f(x) = x²-x+1¹ [0,3] 59. f(t)=t-√√t, [-1,4] 60. f(x) = [0, 3] 61. f(t) = 2 cost + sin 2t, [0, π/2] 62. f(0) = 1 + cos²0, [π/4, π] 63. f(x)=x²lnx, [1,4] 64. f(x) = xex/2, [-3,1] ex 1+x²³ 100 cos²x 10+ x² 65. f(x) = ln(x² + x + 1), [-1, 1] 66. f(x)= x 2 tan¹x, [0, 4] 1 67. If a and b are positive numbers, find the maximum value of f(x) = x(1 − x), 0 ≤ x ≤ 1. 68. Use a graph to estimate the critical numbers of f(x) = 1 + 5x = x³ correct to one decimal place. 69-72 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 69. f(x) = x³ x³ + 2, −1≤ x ≤ 1 70. f(x) = e+ e- ²x, 0≤x≤ 1 71. f(x)=x√√x-x² 72. f(x)= x 2 cos x, -2 ≤x≤0