HW 9 Solutions - Mathematics 408K: Homework 9 Gagan Tara...

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Mathematics 408 K: Homework 9 Gagan Tara Nanda 18 th October, 2004 HW 9 : # s 6 , 12 , 22 6. Make sure you understand the English carefully here: The idea is to maximize the “rate of growth of sales”, which in Math means maximize N 0 ( t ) .Sowew ishtomax im izethefunct ion N 0 ( t ) = 108 t 3 t 2 on the interval 1 t 44 .Soa sa lway s , f nd its derivative. We get N 00 ( t )=108 6 t , which we set to zero to get t =18 . The derivative exists everywhere, so there are no other critical points. We need to then compare the values of N 0 ( t ) at the endpoints and the critical point. First, let’s check that we have a local maximum at t =18 .U s ethe 2 nd Derivative Test. So N 000 ( t )= 6 , which is > 0 for all t .So N 0 (18) is a local maximum. We then have N 0 (1) = 105 , N 0 (44) = 1056 , N 0 (18) = 972 , so the maximum rate of growth of sales occurs on day 18 . 12. Let f ( x )= x 5 +2 x +5
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HW 9 Solutions - Mathematics 408K: Homework 9 Gagan Tara...

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