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Q5Solutions - MAC 2233 Quiz 5 1 Let f(x = 3x4 8x3 Find all...

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MAC 2233 Quiz 5 May 10, 2009 1. Let f ( x ) = 3 x 4 + 8 x 3 . Find all absolute extrema of f on the interval [ - 1 , 1]. Solution: First, find the critical points of f that are in the given interval: f 0 ( x ) = 12 x 3 + 24 x 2 = 12 x 2 ( x + 2) so the critical points are x = 0 and x = - 2. Since x = - 2 is not in the given interval, we can disregard it. The next step is to compare f ( - 1), f (0) and f (1). Since f ( - 1) = - 5, f (0) = 0 and f (1) = 11, the absolute maximum value of f is 11 (which occurs at x = 1 and the absolute minimum value of f is - 5 (which occurs at x = - 1). 2. Let f ( x ) = x x - 1 . It can be shown that f (0) = 0, that f has a horizontal asymptote at y = 1 and that f has a vertical asymptote at x = 1. It can also be shown that lim x 1 - f ( x ) = -∞ , and that lim x 1 + f ( x ) = + . Complete the following steps: (a) Determine the intervals where f is increasing and the intervals where f is decreas- ing. Solution: We want to find where f 0 is positive and where f 0 is negative. A routine application of the quotient rule shows that f 0 ( x ) = - 1 ( x - 1) 2 .
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