HW2Sol - Fall 2009 IEOR 160 Industrial Engineering &...

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Fall 2009 IEOR 160 September 15, 2009 Page 1 of 1 HOMEWORK 2 SOLUTIONS Chapter 12.4 1. 1. Let f ( x ) be the pro±t if $x is spent on advertising. Then f (0) = 0 and for x > 0, f ( x ) = 300 x - 100 x - 5000 - x. Since f ( x ) has no derivative at x = 0, maximum pro±t occurs either for x = 0 or a point where f ( x ) = 0 . Now for x > 0 f ( x ) = 100 x - 1 = 0 for x = 10 , 000. Also f ′′ ( x ) = 50 x 3 / 2 < 0 for x > 0. Thus x = 10 , 000 is a local maximum (and a maximum over all x > 0). We now compare f (0) and f (10 , 000) to determine what the company should do. f (0) = 0 and f (10 , 000) = $5 , 000, so the company should spend $10,000 on advertising. If ±xed cost is $20,000, f ( x ) = 0 still holds for x = 10 , 000. Comparing f (0) = 0 and f (10 , 000) = - 10 , 000, we now ±nd that x = 0 is optimal. 2. Let f ( q ) be the pro±t if q units are produced. Then f (0) = 0 and for q > 0, f ( q ) = q (100 - 4 q ) - 2 q - 50. For
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