# ep8 - 3 Critical numbers 2 1 ± = x Absolute Max 53...

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92.131 Calculus 1 Extreme Values Find all the critical numbers for the following functions, and then determine the absolute maximum and minimum values on the stated interval. 1) 1 2 4 1 ) ( 2 4 = x x x f on [ 1, 3] 2) 1 4 3 1 ) ( 3 = x x x f on [ 3, 1] 3) 5 4 4 ) ( 2 4 + = x x x f on [] 2 , 0 4) t t t g 16 3 1 ) ( 3 + = on [] 3 , 1 5) 4 ) ( 2 + = x x x f on [] 4 , 0 6) 1 4 3 ) ( 2 + = x x x f on [] 1 , 1 7) 3 ) ( 3 2 = x x x f on [] 1 , 1 8) x x x f ln ) ( = on 2 , 1 e e

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92.131 Calculus 1 Extreme Values Solutions: 1) ) 2 )( 2 ( 4 ) ( 3 + = = x x x x x x f , and so critical numbers are 2 , 0 ± = x . Since 2 = x is not in the interval [ −1 , 3] we need not consider it. 4 11 ) 1 ( = f , 1 ) 0 ( = f 5 ) 2 ( = f , and 4 5 ) 3 ( = f . So the absolute maximum value is 4 5 ) 3 ( = f , and the absolute minimum value is 5 ) 2 ( = f . 2) 4 ) ( 2 = x x f , and so critical numbers are 2 ± = x . Since 2 = x is not in the interval [ 3, 1] we need not consider it. 2 ) 3 ( = f , 3 13 ) 2 ( = f , and 3 14 ) 1 ( = f . So the absolute maximum value is 3 13 ) 2 ( = f , and the absolute minimum value is 3 14 ) 1 ( = f
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Unformatted text preview: . 3) Critical numbers: 2 1 , ± = x ; Absolute Max: 53 Absolute Min: 4 4) Critical numbers: 2 , ± = t ; Absolute Max: 3 49 Absolute Min: 3 32 5) 2 2 2 2 2 2 2 2 ) 4 ( ) 2 )( 2 ( ) 4 ( 4 ) 4 ( ) 2 ( ) 4 ( 1 + − + = + − = + ⋅ − + ⋅ = ′ x x x x x x x x x f so critical numbers are 2 ± = x Since x = − 2 is not in the interval [0, 4], we do not consider it. ) ( = f , 4 1 ) 2 ( = f , and 5 1 ) 4 ( = f . The absolute min is 0, and the absolute max is 4 1 . 6) No Critical numbers > ′ f ; Absolute Max: 5 3 Absolute Min: 5 3 − 7) Critical numbers: 3 3 6 , 3 , − = x ; Absolute Max: 0 Absolute Min: 2 1 − 8) Critical numbers: 1 , = x ; Absolute Max: 2 2 − e Absolute Min: 1...
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## This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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ep8 - 3 Critical numbers 2 1 ± = x Absolute Max 53...

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