2.9 Practice Prelim Answers - Solutions to Review Problems...

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Solutions to Review Problems for Prelim 2 Solutions to Maxima, minima, convexity, concavity 1. f ( x )= x 3 +3 x 2 - 9 x +6, f ± ( x )=3 x 2 +6 x - 9, f ±± ( x )=6 x +6. (a) 0 = 3( x 2 +2 x - 3) = 3( x + 3)( x - 1) when x =1 , x = - 3. (b) f ±± (1) = 12 > 0, local min., f ±± ( - 3) = - 12 < 0, local max. (c) increasing when x< - 3, x> 1, (d) convex when x> - 1. 2. f ( x )= x 3 / 3 - 5 x 2 / 2+4 x , f ± ( x )= x 2 - 5 x +4, f ±± ( x )=2 x - 5. (a) 0 = x 2 - 5 x +4=( x - 1)( x - 4) when x =1 , x =4 . (b) f ±± (4) = 3 > 0, local min., f ±± (1) = - 3 < 0, local max. (c) increasing when x< 1, x> 4, (d) convex when x> 5 / 2. 3. f ( x )= x - 6 x +2ln x , f ± ( x )=1 - 3 x - 1 / 2 +2 x - 1 , f ±± ( x )=(3 / 2) x - 3 / 2 - 2 x - 2 . (a) 0 = 1 - 3 x - 1 / 2 - 2 x - 1 =(1 - 2 x - 1 / 2 )(1 - x - 1 / 2 ) when x =1, x =2. (b) f ±± (1) = 3 / 2 - 2 < 0, local max., f ±± (4) = 3 / 16 - 2 / 16 > 0, local min. (c) increasing when 0 <x< 1, x> 2 (d) convex when x 1 / 2 > 4 / 3. 4. f ( x )= x 8 / 5 + x - 2 / 5 , f ± ( x )=(8 / 5) x 3 / 5 - (2 / 5) x - 7 / 5 , f ±± ( x )=(24 / 25) x - 2 / 5 +(14 / 25) x - 12 / 5 (a) 0 = (8 / 5) x 3 / 5 - (2 / 5) x - 7 / 5 when x 2 =1 / 4or x =1 / 2 (b) f ±± ( x ) > 0 for all
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This note was uploaded on 11/14/2007 for the course MATH 1106 taught by Professor Durrett during the Spring '07 term at Cornell University (Engineering School).

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2.9 Practice Prelim Answers - Solutions to Review Problems...

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