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m116q08a - 1 c [email protected] MATHA116 Survey of Calculus...

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1 MATHA116, Survey of Calculus, Quiz #8 Solutions Updated on November 7, 2006 Get help whenever in doubt! c [email protected] D irection. X-1. Please read each of the following questions carefully. Show your work as much as possible. No partial credit will be given without proper work shown. Write your answer above the Answer line. Use the back of each page should more space be needed. N Important: Direction. Q uestion X-1. ( 40 points ) Find the relative extreme value , if any, of the function f ( t ) = t 2 - 16 t . W ork X-1. We’ll find the extreme values following the procedure as shown below. (1) Find all critical numbers. f ( t ) = t 2 - 16 t = t 2 - 16 t - 1 (1) f 0 ( t ) = 2 t + 16 t - 2 = 2 t + 16 t 2 = 2 t 3 + 16 t 2 (2) f 0 ( t ) = 0 = 2 t 3 + 16 = 0 , 2 t 3 = - 16 , t 3 = - 8 , t = - 2 (3) indicating that - 2 is a critical number of the function. (2) Use the 2nd derivative test to identify the extreme value. f 0 ( t ) = 2 t + 16 t - 2 (4) f 00 ( t ) = 2 + 16( - 2) t - 3 = 2 - 32 t 3 (5) f 00 ( - 2) = 2 - 32 ( - 2) 3 = 2 + 4 = 6 > 0 (6) We therefore conclude that f ( - 2) = ( - 2) 2 - 16 ( - 2) = 4+8 = 12 is a relative minimum value. The answer is the function has a relative minimum value of 12, but has no relative maximum value.
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