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please do 36 , 46 and 61
please do 36 , 46 and 61
In Problems 5-8, the derivative of a continuous function f is given. Find the open intervals on which f is (a) increasing; (b) decreasing; and (c) find the x-values of all relative extrema. 5. f'(x) = (x+3)(x - 1)(x-2) 6. f'(x) = 2x(x - 1)³ 7. f'(x) = (x + 1)(x - 3)² 8. f'(x) = In Problems 9-52, determine where the function is (a) increasing; (b) decreasing; and (c) determine where relative extrema occur. Do not sketch the graph. 9. y=-³-1 11. y=x-x²+2 13 - - 2 + 5x − 2 13. y=3 15. y=x²-2² 17. y=-²+ 31. y= 33. y= 25. y = r — 5r +200 27. y 8-18 29. y = (x²-4)4 5 35. y = 37. y = 39. y= ²+2x-5 19 19. y=2³- -2x²+10x+2 20. y = −5x³ + x² + x −7 9 5 21. y=3 23. y=35-5.r³ 47 22. y=-³-² + 10x 3 16 24. y 3x- (Remark: x + 2 ³+²+x+1=0 has no real roots.) 3x4 2 - 5x² +22x + 1 10 √x 1² 2-x 1²-3 x+2 ar² + b cx²+d 10. y=x² + 4x +3 5 12. y=x²- -2x+6 (a) for ad-be > 0 (b) for ad-bc < 0 41. y=(x-1)2/3 43. y=x²(x-6)* 45. ye+ π 47. y=x²-9 In x 49. y=e-e-* x+ 14. y=4 x(x+2) x²+1 16. y=-3+12r-r³ 18. y=x¹6x² + 12x-6 26. y= -r³ 32. y= 4 28. y= 5- 34. y = -4x² +17 13 30. y = x(x-2) 3 x ax + b cx+d (a) for ad-bc> 0 (b) for ad-bc0 1 38. y= 36. y = 4x² + 21² 4x2-25 for d/c<0 40. y = √√x³9x +3x+4 42. y = x²(x+3)4 44. y=(1-x)2/3 46. y=xlnx 48. y=xle 50. y=e-²¹/2 Section 13.1 Relative Extrema 587 51. y=xlnx-r 52. y= (²+1)e-* In Problems 53-64, determine intervals on which the function is increasing; intervals on which the function is decreasing; relative extrema; symmetry; and those intercepts that can be obtained conveniently. Then sketch the graph. 53. y=x²-3x - 10 55, y = 3x - 57. y=2r³-9r² + 12x 59. y=-21² 61. y(x-1)²(x + 2)² 63. y = 2√√x-x 54. y=2x²+x-10 56. y=x²-16 58. y=2¹-²-4x+4 60. y=x6- 62. y=√(x²-x-2) 64. y=x³/3-212/3 65. Sketch the graph of a continuous function f such that f(2)=2, f(4)=6, f'(2) = f'(4) = 0, f'(x) < 0 for x < 2, f'(x) > 0 for 2 < x < 4,f has a relative maximum at 4, and lim-f(x) = 0. 66. Sketch the graph of a continuous function f such that f(1) = 2, f(4)=5, f'(1) = 0, f'(x) ≥ 0 for x < 4, f has a relative maximum when x = 4, and there is a vertical tangent line when x = 4. کی 67. Average Cost If cf = 25,000 is a fixed-cost function, show that the average fixed-cost function cf = c/q is a decreasing function for q> 0. Thus, as output q increases, each unit's portion of fixed cost declines. 68. Marginal Cost If c=3q-3q² + q³ is a cost function, when is marginal cost increasing? 69. Marginal Revenue Given the demand function p=500-5q find when marginal revenue is increasing.. 70. Cost Function For the cost function c= √q, show that marginal and average costs are always decreasing for q> 0. 71. Revenue For a manufacturer's product, the revenue function is given by r = 240g +57q²q³. Determine the output for maximum revenue. 72. Labor Markets Eswaran and Kotwal' consider agrarian economies in which there are two types of workers, permanent and casual. Permanent workers are employed on long-term contracts and may receive benefits such as holiday gifts and emergency aid. Casual workers are hired on a daily basis and perform routine and menial tasks such as weeding, harvesting, and threshing. The difference z in the present-value cost of hiring a permanent worker over that of hiring a casual worker is given by z = (1+b)wp-bwe where w, and we are wage rates for permanent labor and casual labor, respectively, b is a positive constant, and w, is a function of We. (a) Show that dz = (1+b) dwe dwp dwe b 1+b (b) If dwp/dwe<b/(1+b), show that z is a decreasing function of we. M. Eswaran and A. Kotwal, "A Theory of Two-Tier Labor Markets in Agrarian Economics," The American Economic Review, 75, no. 1 (1985), 162-77.
In Problems 5-8, the derivative of a continuous function f is given. Find the open intervals on which f is (a) increasing; (b) decreasing; and (c) find the x-values of all relative extrema. 5. f'(x) = (x+3)(x - 1)(x-2) 6. f'(x) = 2x(x - 1)³ 7. f'(x) = (x + 1)(x - 3)² 8. f'(x) = In Problems 9-52, determine where the function is (a) increasing; (b) decreasing; and (c) determine where relative extrema occur. Do not sketch the graph. 9. y=-³-1 11. y=x-x²+2 13 - - 2 + 5x − 2 13. y=3 15. y=x²-2² 17. y=-²+ 31. y= 33. y= 25. y = r — 5r +200 27. y 8-18 29. y = (x²-4)4 5 35. y = 37. y = 39. y= ²+2x-5 19 19. y=2³- -2x²+10x+2 20. y = −5x³ + x² + x −7 9 5 21. y=3 23. y=35-5.r³ 47 22. y=-³-² + 10x 3 16 24. y 3x- (Remark: x + 2 ³+²+x+1=0 has no real roots.) 3x4 2 - 5x² +22x + 1 10 √x 1² 2-x 1²-3 x+2 ar² + b cx²+d 10. y=x² + 4x +3 5 12. y=x²- -2x+6 (a) for ad-be > 0 (b) for ad-bc < 0 41. y=(x-1)2/3 43. y=x²(x-6)* 45. ye+ π 47. y=x²-9 In x 49. y=e-e-* x+ 14. y=4 x(x+2) x²+1 16. y=-3+12r-r³ 18. y=x¹6x² + 12x-6 26. y= -r³ 32. y= 4 28. y= 5- 34. y = -4x² +17 13 30. y = x(x-2) 3 x ax + b cx+d (a) for ad-bc> 0 (b) for ad-bc0 1 38. y= 36. y = 4x² + 21² 4x2-25 for d/c<0 40. y = √√x³9x +3x+4 42. y = x²(x+3)4 44. y=(1-x)2/3 46. y=xlnx 48. y=xle 50. y=e-²¹/2 Section 13.1 Relative Extrema 587 51. y=xlnx-r 52. y= (²+1)e-* In Problems 53-64, determine intervals on which the function is increasing; intervals on which the function is decreasing; relative extrema; symmetry; and those intercepts that can be obtained conveniently. Then sketch the graph. 53. y=x²-3x - 10 55, y = 3x - 57. y=2r³-9r² + 12x 59. y=-21² 61. y(x-1)²(x + 2)² 63. y = 2√√x-x 54. y=2x²+x-10 56. y=x²-16 58. y=2¹-²-4x+4 60. y=x6- 62. y=√(x²-x-2) 64. y=x³/3-212/3 65. Sketch the graph of a continuous function f such that f(2)=2, f(4)=6, f'(2) = f'(4) = 0, f'(x) < 0 for x < 2, f'(x) > 0 for 2 < x < 4,f has a relative maximum at 4, and lim-f(x) = 0. 66. Sketch the graph of a continuous function f such that f(1) = 2, f(4)=5, f'(1) = 0, f'(x) ≥ 0 for x < 4, f has a relative maximum when x = 4, and there is a vertical tangent line when x = 4. کی 67. Average Cost If cf = 25,000 is a fixed-cost function, show that the average fixed-cost function cf = c/q is a decreasing function for q> 0. Thus, as output q increases, each unit's portion of fixed cost declines. 68. Marginal Cost If c=3q-3q² + q³ is a cost function, when is marginal cost increasing? 69. Marginal Revenue Given the demand function p=500-5q find when marginal revenue is increasing.. 70. Cost Function For the cost function c= √q, show that marginal and average costs are always decreasing for q> 0. 71. Revenue For a manufacturer's product, the revenue function is given by r = 240g +57q²q³. Determine the output for maximum revenue. 72. Labor Markets Eswaran and Kotwal' consider agrarian economies in which there are two types of workers, permanent and casual. Permanent workers are employed on long-term contracts and may receive benefits such as holiday gifts and emergency aid. Casual workers are hired on a daily basis and perform routine and menial tasks such as weeding, harvesting, and threshing. The difference z in the present-value cost of hiring a permanent worker over that of hiring a casual worker is given by z = (1+b)wp-bwe where w, and we are wage rates for permanent labor and casual labor, respectively, b is a positive constant, and w, is a function of We. (a) Show that dz = (1+b) dwe dwp dwe b 1+b (b) If dwp/dwe<b/(1+b), show that z is a decreasing function of we. M. Eswaran and A. Kotwal, "A Theory of Two-Tier Labor Markets in Agrarian Economics," The American Economic Review, 75, no. 1 (1985), 162-77.