Please help me and show work Only Numbers: 1, 7, 9, 21, 31, 37, 42

302 CHAPTER 5 Graphs and the Derivative 35. x} = X2 + 10x 9 37. r) = I'_)r3 + 9x3 + 168x 3 38. f(x) = .r3 12x2 45x + 2 36. fit} = 8 6x x3 3 72 39. f('() = E 40. f(x) = x + l 41. f(\\'} = _t(x + 5)2 42. fit} = x(x 3}2 43. x) 7 18x 187. -' 44. fix) 7 26 45. f(x) = x3\" 42:5'3 46. x) = 3;?\" + 5614'8 47. x) 7 ln(.r3 + 1} 4s. x) 7 .72 + 3 ln Ix + 1' 49. x} 7.81%; .r 50. x} 7 5 For each of the exercises listed below, suppose that the function that is graphed is not x), but f'(x). Find the open intervals where the original function is concave upward or concave downward, and find the location of any inection points. 51. Exercise 29 53. Exercise 31 52. Exercise 30 54. Exercise 32 55. Give an example of a function fix) such that _f'(0} = 0 but Fm) does not exist. Is there a relative minimum or maximum or an inection point at x = 0? 56. (3) Graph the two functions it) = x7\" and g(x) = its\" on the window _-2, 2] by [-2, 2]. (1)} Verify that both f and 5; have an inection point at (0. O}. (c) How is the value of If\"(0) different from g\"()? Q .. (d) Based on what you have seen so far in this exercise, is it always possible to tell the difference between a point where the second derivative is 0 or undened based on the graph? Explain. 57. Describe the slope ofthe tangent line to the graph ofx) = c" for the following. (a) 1% -x (ll) x70 58. What is true about the slope of the tangent line to the graph of fix) = In); as x) f5? As x* 0'? Find any critical numbers for f in Exercises 5966 and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f\"(c) = 0 or f'(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. 59. x) 7 73:2 7 101' 7 25 60. fix) 7 x2 7 121- + 36 (.1. x} 7 3x3 3x: + l 63. x) = (x + 3t)\" 65. m) 7 x713 + x4\" (.2. x) 7 213 4x: + 2 64. t} = x3 66. at} 7 .6\" + x5\" Sometimes the derivative of a function is known, but not the function. We will see more of this later in the book. For each function f ' dened in Exercises 6770, nd f"(x), then use a graphing calculator to graph f' and f " in the indicated window. Use the graph to do the following. (a) Give the (approximate) x-values where f has a maximum or minimum. (b) By considering the sign of fix). give the (approximate) inter- vals where x} is increasing and decreasing. (c) Give the (approximate) xvalues of any inection points. (d) By considering the sign of _f"[.r}. give the intervals where f is concave upward or concave downward. 67. f'(x} 2 x3 - 6x3 + "Li + 4; [-5,5] by :5. 15] 68. f'(x} = 10.r2(x l}(5x 3); _-l, 1.5] by [20. 20] 7 1 x: . (x: + 112' 70. f'()() 7 .8 + x lnx; [0, 1] by'72, 2] 71. Suppose a friend makes the following argument. A function f is increasing and concave downward. Therefore, I)\" is positive and decreasing, so it eventually becomes 0 and then negative. at which point f decreases. Show that your friend is wrong by giving an example of a function that is always increasing and concave downward. 6'). fix) _-3. 3] by [-15, 1.5] APPLICATIONS Business and Economics Point of Diminishing Returns In Exercises 7275, nd the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). 72. at.) 7 10,000 .{3 + 42x: + 800x, 0 5 ,1' 1: 20 4 3 . . 73. stx) 7 27- (7x- + 66x- + 10507 7 400), 0 5 .1' 5 25 74. stx) 7 70.3.73 7 x3 + 11.4.r, 0 5 x 5 6 75. RM) = 416.9 7* 3.7xE + 5.1', 0 5 .r 6 i A 76. Risk Aversion ln economics. an index of absoittte risk aver- sion is dened as where M measures how much of a commodity is owned and UfM) is a utiiig'function, which measures the ability of quan- tity M of a comm_odity to satisfy a consumer's wants. Find If M ) for MM} 2 VM and for UfM) 2 M3\5.3 Higher Derivatives, Concavity, and the Second Derivative Test 301 Test a number in the interval (0, 200) to see that R"(x) is positive there. Then test a num- ber in the interval (200, 600) to find that R"(x) is negative in that interval. Since the sign of R"(x) changes from positive to negative at x = 200, the graph changes from concave upward to concave downward at that point, and there is a point of diminishing returns at the inflection point (200, 10665). Investments in advertising beyond $200,000 return less and less for each dollar invested. Verify that R' (200) = 8. This means that when $200,000 is invested, another $1000 invested returns approximately $8000 in additional revenue. Thus it may still be eco- nomically sound to invest in advertising beyond the point of diminishing returns. 5.3 WARM-UP EXERCISES Find all relative extrema for the following functions, as well as the x-values where they occur. (Sec. 5.2) W1. f(x) = x3 - 3x2 - 72x + 20 W2. f (x) = x4 - 8x3 - 32x2 + 10 5.3 EXERCISES Find f"(x) for each function. Then find f"(0) and f"(2). 26. Let f(x) = In x. 1 . f (x) = 5x3 - 7x2 + 4x + 3 (a) Compute f' (x), f"(x), f" (x), f(+(x), and f(5) (x). 2. f (x) = 4x3 + 5x2 + 6x - 7 (b) Guess a formula for f("(x), where n is any positive integer. 3. f (x) = 4x4 - 3x3 - 2x2 + 6 27. For f(x) = el, find f"(x) and f"(x). What is the nth derivative of f with respect to x? 4 . f (x ) = -x4+7x3 - 2 28. For f(x) = a", find f"(x) and f"(x). What is the nth derivative of f with respect to x? 5. f(x) = 3x2 - 4x + 8 6. f(x) = 8x2 + 6x + 5 7. f ( x) = - 8. f ( x ) = * In Exercises 29-50, find the open intervals where the functions 1 - x2 are concave upward or concave downward. Find any inflection points. 9. f (x) = Vx + 4 10. f(x) = V2x2+9 29. 30. 11. f(x) = 32x3/4 12. f(x) = -6x1/3 13. f(x) = 5ex 14. f(x) = 0.5er 15. f(x) = In x (3, 7) 4x 16. f (x) = Inx + (2, 3) Find f"(x), the third derivative of f, and f()(x), the fourth 3- derivative of f, for each function. - 2 of 17. f(x) = 7x4 + 6x3 + 5x2 + 4x+ 3 + + 18 . f (x ) = - 2x4+ 7x3 + 4x2 + x 19. f(x) = 5x5 - 3x4 + 2x3 + 7x2 + 4 31. 32. 20. f(x) = 2x5 + 3x4 - 5x3 + 9x - 2 21. f(x) = * - 1 22. f (x ) = * + 1 x + 2 23. f (x ) = - 3x (8, 6 x - 2 24. f ( x ) = = (-1, 7) -8 2x + 1 (6, -1) 25. Let f be an nth degree polynomial of the form f(x) = x" + an 1x" + ... + ax + do. -2 10' 8 (-2, -4) (a) Find f(") (x). (b) Find f )(x) for k > n