587 Relative Extrema Section 13.1 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. 1) -2) 6. S(x) = 2r(x - 1) %3D 54. y = 2x +x-10 56. y =x- 16 58. y = 2x - ²-4x +4 3+x) = (x),/ 2 8. S'(x) = 53. y=x- 3x 10 55. y 3x- 57. y 2r'-9x+ 12x 60. y = * - r (x) are clude 10. y =x+ 4x +3 59. y= x-2 9.y=ード-1 61. y = (x- 1) x+2) 63. y = 2x- x 62. y = (x2 -x- 2) also 11. y=x – x² +2 - 2r +6 64. y = x/3 – 2x3/3 | 65. Sketch the graph of a continuous function f such that S(2)= 2, f(4) = 6, f'(2) =f'(4) = 0, f'(x) < 0 for x < 2, f'(x) > 0 for 2 0 for x < 4, ƒ has a relative maximum when x = 4, and there is a vertical tangent line when x = 4. e the 13. y=-3 - 2r + 5x - 2 14. y = 4. 16. y =-3 + 12x – x use || 15. y=x- 2r² 7. 17. y=x-パ+ 2r-5 18. y = x – 6x? + 12x – 6 | 61 x²+ 10x + 2 20. y = -5x³ +x² + x – 7 19. y= 2r3 – - 5x² + 22x + 1 6. 22. y = 47 If cf = 25,000 is a fixed-cost function, show 67. Average Cost that the average fixed-cost function f = cf/q is a decreasing function for q > 0. Thus, as output q increases, each unit's portion of fixed cost declines. 21. y = 3. %3| %3D 3 9-r (Remark: x + 2 23. y = 3x - 5x³ 24. y = 3x - x' + x² + x +1 = 0 has no real roots.) If c = 39 – 3q² +q° is a cost function, 68. Marginal Cost when is marginal cost increasing? - 3x4 Given the demand function 25. y =-x – 5xª + 200 26. y = - 4x3 + 17 2 69. Marginal Revenue p = 500 – 5q 4. 28. y = 13 27. y = 8x - x8 + 3x +4 %3D 3 find when marginal revenue is increasing. 30. y = x(x – 2) q, show that 29. y= (r2 – 4)4 For the cost function c = 70. Cost Function fxl marginal and average costs are always decreasing for q > 0. For a manufacturer's product, the revenue %3D - 3 32. y = 31. y = 71. Revenue function is given by r = 240q +57q² – q³. Dete for maximum revenue. rmine the output 33. y=Tx 34. y = 9+ xp %3D 72. Labor Markets Eswaran and Kotwal' consider agrarian (a) for ad – bc > 0 (b) for ad – bc < 0 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 36. y = 4x2 + 35. y = %3D x² - 3 2r2 37. y = 38. y = 4x2 - 25 %3D x+ 2 | 39. y= for d/c < 0 40. y = Vr – 9x z = (1+ b)wp – bwc - - %3D %3D (a) for ad - bc > 0 (b) for ad - bc < 0 pane Wc are wage rates for permanent labor and casual where wp labor, respectively, b is a positive constant, and w, is a function of wc. 42. y = x²(x+3)4 (a) Show that 43. y = r'(r-6) 44. y = (1 – x)2/3 zp = (1+b) dwp [dwc 46. y x In x %3D dwe (b) If dw,/dw. < b/(1+b), show that z is a decreasing functic of Wc. 47. y = x-9 In x 48. y =x-le 49. y = e-e-r 50. y = e2 'M. Eswaran and A. Kotwal, "A Theory of Two-Tier Labor Markets in Ag Economics," The American Economic Review, 75, no. 1 (1985), 162-77,