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4.6 Rates of Change in Business and Economics 26. A soft-drink company estimates that the cost, C, in dollars of producing x litres of product per day is C(x) 0.0005x 2 4x 4000. (a) Find the marginal cost if 5000 L is produced. (b) Determine the daily production level to minimize the average cost. 27. A publisher can sell x thousand copies of a monthly sports magazine if it x sells for p 5 dollars. The monthly cost of publishing, C, can be 10 0 modelled by C(x) 8000 200x 0.05x 2 . (a) Determine the revenue function. (b) Determine the profit function. (c) Calculate the profit on the sale of 20 000 magazines. (d) Calculate the marginal profit on the sale of 30 000 magazines. (e) Calculate the maximum profit and the sales required to realize the profit. 28. The cost to produce one high-definition TV is $750. The manufacturer charges $1800 per unit for orders of 50 or less. The price will be reduced by $15 per unit for larger orders. Determine the size of order that (a) maximizes revenue (b) maximizes profit 4.7 Sketching Graphs of Polynomial Functions: Concavity 29. Determine the intervals of concavity and the points of inflection. (a) y x 3 3x 2 45x (b) y 2x 3 9x 2 108x 200 (c) y x 4 54x 2 250 (d) y x 4 12x 2 30. Determine the intervals where each function increases or decreases. Also determine where the graph of the function is concave up or down, the maximum and minimum points, and the points of inflection. Use the information to graph each function. (a) y 2x 2 12x 8 (b) y 2x 3 3x 2 12x 3 (c) y 2x 3 3x 2 36x (d) y x 4 32x (e) y x 3 x 2 5x 3 (f) y 3x 5 5x 3 60x 31. Use the second derivative test to show that f (x) x 3 2x 2 has a local maximum at the origin. REVIEW QUESTIONS BY ACHIEVEMENT CHART CATEGORIES Knowledge and Understanding 32. (a) Determine the intervals where f (x) x 4 4x 3 4x 2 1 is increasing and decreasing. (b) Identify all maximum and minimum values. CHAPTER 4 REVIEW 337
33. (a) Find the intervals of concavity and all inflection points for the graph of f (x) x 3 6x 2 9x 3. (b) Do you need any more information to graph f ? If so, what would be this information? 34. Determine the level of production that will maximize profits if the cost, C, in dollars of producing x units in thousands is C(x) x 2 4x 100 8 and the selling price is p 80 x . 2 Communication 35. Explain why the following is false: Let c be a critical number of f (x). Then f (x) has a local maximum or minimum at f (c). 36. f ′(x) < 0 for all x in the interval 3 ≤ x ≤ 10. Explain why f (3) > f (6). 37. For function f (x), f ′(x) is decreasing on an interval. Graph f (x) where (a) f ′(x) < 0 (b) f ′(x) > 0 Application 38. An object moves horizontally along a line. Its position from a fixed point can be modelled by s(t) t 3 6t 2 15t 20, where s is the displacement in centimetres and t is the time in seconds. Determine the object’s maximum velocity on the interval 0 ≤ t ≤ 5. 39. A cylindrical, open container is to be made to hold 2 L (2000 cm 3 ) of liquid. The material for the bottom costs $0.05/cm 2 . The material for the sides costs $0.10/cm 2 . What are the dimensions of the least expensive container? 40. A toy manufacturer has determined that the total cost, C, of operating a factory is C(x) 0.5x 2 45x 10 000, where x is the number of units produced in thousands. Determine the production level that minimizes the average cost. Thinking, Inquiry, Problem Solving 41. Graph function f (x) so that • points (1, 10) and (3, 1) are local extrema on the graph • (1, 3) is an inflection point • the graph is concave down only when x < 1 • the x-intercept is 4 and the y-intercept is 8 42. Prove that an nth-degree polynomial, f (x) a 0 x n a 1 x n 1 … a n , a 0 ≠ 0, has at most n 2 inflection points. 43. Ron is planting a garden all around a rectangular patio. The garden will be 5 m wide, except at the diagonals. The area of the patio must be 150 m 2 . Find the overall dimensions of the garden and patio in which the area of the garden is a minimum. 338 CHAPTER 4 USING THE DERIVATIVE TO ANALYZE POLYNOMIAL FUNCTION MODELS
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