B find the x values where f x has a local minimum or

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b ) Find the x values where f ( x ) has a local minimum or maximum. Solution: Applying the first derivative test, we can pick specific points to try. f ( - 1) = - 16, so f 0 ( x ) is negative on ( -∞ , 0). f (2) = - 16, so f 0 ( x ) is negative on (0 , 3). f (4) = 64, so f 0 ( x ) is positive on (3 , ). It follows from the above that f ( x ) has a local minimum at x = 3 (and an inflection point at x = 0). c ) Find the intervals where f ( x ) is concave up and where it is concave down. Solution: f 00 ( x ) = 12 x 2 - 24 x = 12 x ( x - 2) 5
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Applying the test for inflection points, we see that f ( - 1) = 36, so f 00 ( x ) is positive on ( , 0). f (1) = - 12, so f 00 ( x ) is negative on (0 , 2). f (3) = 36, so f 00 ( x ) is positive on (2 , ). It follows from the above that f ( x ) is concave up on ( , 0) (2 , ) and concave down on (0 , 2). d ) Find points of inflection for f ( x ). Solution: From part c ), since f 00 ( x ) changes sign at x = 0 , 2, both (0 , 0) and (2 , - 16) are points of inflection for f ( x ). 6
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4. (10 points) A car is traveling 84 ft/sec begins to decelerate at a constant rate of 14 ft/sec 2 . After how many seconds does the car come to a stop and how far will the car have traveled before stopping? Solution: We have v (0) = 84, a ( t ) = v 0 ( t ) = - 14, so by the Fundamental Theorem of Calculus, v ( x ) - v (0) = Z x 0 v 0 ( t ) dt = Z x 0 - 14 dt = [ - 14 t ] x 0 = - 14 x so v ( t ) = - 14 t + 84. We wish to find the time at which the car comes to rest ( v ( t ) = 0), and solving this, we get t = 84 14 = 6. To determine how far the car has traveled before stopping, we want to find the distance traveled by the car between t = 0 and t = 6. This is given by d = Z 6 0 | v ( t ) | dt = Z 6 0 | - 14 t + 84 | dt = Z 6 0 - 14 + 84 dt (since - 14 t + 84 > 0 for 0 < t < 6) = £ - 7 t 2 + 84 t / 6 0 = - 252 + 504 = 252 7
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The car travels 252 feet before stopping. 8
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5. (10 points) A 6 ft tall man walks away from a 15 ft lamp post at a speed of 3 ft/sec. At what rate is the length of his shadow increasing?
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