# test3a - MAC 2233/0001-006 Business Calculus Fall 2011 TEST...

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MAC 2233/0001-006 Business Calculus, Fall 2011 TEST 3 Name: Date: 11/22/2011 Time allowed: 50 minutes Section: Show ALL steps. Fifty points equal 100% (1 point) Question 1 . (4+4 points) Company RMK wants to investigate the e±ect of advertisement on the revenue of a certain product. The following function is used to model the situation: R = - 1 12 x 3 + 2 x 2 + 4 x + 50 , 0 x 15 . where R is the monthly revenue in thousand dollars, and x is the amount of advertising in thousand dollars each month. (a) Determine the point of diminishing returns. Solution. The point of diminishing returns is a point of re²ection. Thus, we use calculus and ³nd that dR dx = - 1 4 x 2 + 4 x + 4 , and d 2 R dx 2 = - 1 2 x + 4 . Candidate(s) for point(s) of re²ection occurs when d 2 R dx 2 = 0 (or when d 2 R dx 2 is unde³ned, which is not possible in this problem). Hence, we have - 1 2 x + 4 = 0, giving that x = 8. To check that we indeed have a point of re²ection when x = 8, we set up the following table: Interval (0 , 8) (8 , 15) Test value 1 10 Sign of d 2 R dx 2 + - Concavity up down We see that concavity changes at x = 8. Thus, the point of diminishing returns occurs at x = 8 thousand dollars, with R = 167 1 3 thousand dollars . (b) Explain the signi³cance of the point of diminishing returns in the company’s decision on allocating advertising dollars. Solution. The point of diminishing returns is the point where the marginal dR dx starts to decrease. If x < 8 (before the point of diminishing returns), it is usually a good idea to consider increasing investment (in advertising dollars in this case) because the marginal return dR dx is increasing as x increases (so that additional investments result in an increasing rate of return). The company should

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test3a - MAC 2233/0001-006 Business Calculus Fall 2011 TEST...

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