Pre-Calc Homework Solutions 160

Pre-Calc Homework Solutions 160 - 160 Section 4.4 28(a We...

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24. (a) f 9 ( x ) is a quadratic polynomial, and as such it can have 0, 1, or 2 zeros. If it has 0 or 1 zeros, then its sign never changes, so f ( x ) has no local extrema. If f 9 ( x ) has 2 zeros, then its sign changes twice, and f ( x ) has 2 local extrema at those points. (b) Possible answers: No local extrema: y 5 x 3 ; 2 local extrema: y 5 x 3 2 3 x 25. Let x be the length in inches of each edge of the square end, and let y be the length of the box. Then we require 4 x 1 y # 108. Since our goal is to maximize volume, we assume 4 x 1 y 5 108 and so y 5 108 2 4 x . The volume is V ( x ) 5 x 2 (108 2 4 x ) 5 108 x 2 2 4 x 3 , where 0 , x , 27. Then V 95 216 x 2 12 x 2 52 12 x ( x 2 18), so the critical point occurs at x 5 18 in. Since V 9 ( x ) . 0 for 0 , x , 18 and V 9 ( x ) , 0 for 18 , x , 27, the critical point corresponds to the maximum volume. The dimensions of the box with the largest possible volume are 18 in. by 18 in. by 36 in. 26.
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