5even - 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 2. (b, c);...

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136 CHAPTER 5 GRAPHING AND OPTIMIZATION 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 2. ( b, c ); ( c, d ); ( f, g ) 4. ( a, b ); ( d, f ); ( g, h ) 6. x = b, g 8. = d, g 10. f has a local maximum at = d , and a local minimum at = b ; does not have a local extremum at = a or at = c . 12. (b) 14. (h) 16. (g) 18. (a) 20. ( ) = -3 2 - 12 '( ) = -6 - 12 ' is continuous for all and '( ) = -6 - 12 = 0 = -2 Thus, = -2 is a partition number for ': Next we construct a sign chart for '. Decreasing Increasing + + + + - - - - -3 -2 0 '( ) ( ) Test Numbers '( ) 0 ! 12( ! ) ! 3 6( + ) Therefore, is increasing on (- ! , -2) and decreasing on (-2, ! ); has a local maximum at = -2, which is (-2) = 12. 22. ( ) = -3 2 + 12 - 5 '( ) = -6 + 12 ' is continuous for all and '( ) = -6 + 12 = 0 = 2 Thus, = 2 is a partition number for '. Next we construct a sign chart for '. '( ) ( ) + + + - - - 0 2 3 Test Numbers ) 0 12( + ) 3 ! 6( ! ) Therefore, is increasing on (- ! , 2) and increasing on (2, ! ); has a local maximum at = 2. 24. ( ) = - 3 - 4 + 8 '( ) = -3 2 - 4 ' is continuous for all and '( ) = -(3 2 + 4) < 0. Thus, is decreasing for all ; no local extrema.
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EXERCISE 5-1 137 26. f ( x ) = -2 3 + 3 2 + 120 '( ) = -6 2 + 6 + 120 which is continuous for all . '( ) = -6( 2 - - 20) = -6( + 4)( - 5) = 0 = -4, 5 Thus, = -4 and = 5 are partition numbers for '. Next, we construct a sign chart for ': - - - + + + + + + + + - - - Increasing Decreasing '( ) ( ) -5 -4 0 5 6 Test Numbers '( ) ! 5 ! 60( ! ) 0 120( + ) 6 ! 60( ! ) Therefore, is decreasing on (- ! , -4) and (5, ! ); increasing on (-4, 5); has a local minimum at = -4 and a local maximum at = 5. 28. ( ) = 4 + 2 3 + 5 '( ) = 4 3 + 6 2 which is continuous for all . '( ) = 4 3 + 6 2 = 2 2 (2 + 3) = 0 = - 3 2 , 0 Thus, = - 3 2 and = 0 are partition numbers for '. Next, we construct a sign chart for : '( ) ( ) -2 -1 0 1 3 2 - Incr. Increasing - - - + + + + + Test Numbers ) ! 2 ! 8( ! ) ! 1 2( + ) 1 10( + ) Therefore, is decreasing on !" , ! 3 2 # $ % & ; increasing on ! 3 2 , " # $ % & ; (-1.5) = 3.3125 is a local minimum. 30. ( ) = ln x – x , > 0 f’ ( ) = (1)ln + 1 " # $ % & - 1 = ln + 1 – 1 = ln ( ) = ln = 0 for = 1. ( ) < 0 for < 1 and ( ) > 0 for > 1. Therefore, is decreasing on (0, 1) and increasing on (1, ! ). has a local minimum at = 1. 32. ( ) = ( 2 – 9) 2/3 ( ) = 2 3 ( 2 – 9) -1/3 (2 ) = 3( 2 " 9) ( ) = 0 at = 0. We have to determine the sign of ( ) in the intervals (- ! , -3), (-3, 0) and (0, ! ). For < -3, 2 > 9 and hence ( 2 – 9) 1/3 > 0. Thus, ( ) < 0 on the interval (- ! , -3).
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138 CHAPTER 5 GRAPHING AND OPTIMIZATION For -3 < x < 0, 9 > 2 > 0 and hence ( 2 – 9) 1/3 < 0. Thus, f ’( ) > 0 on the interval (-3, 0). For 0 < < 3, 0 < 2 < 9 and hence ( 2 – 9) 1/3 < 0. Thus, f’ ( ) < 0 on the interval (0, 3). For > 3, 2 > 9 and hence ( 2 – 9) 1/3 > 0. Thus, ( ) > 0 on the interval (3, ! ). Summary: is decreasing on (- ! , -3) and (0, 3); increasing on (-3, 0) and (3, ! ). has a local maximum at = 0 and local minima at = -3 and = 3. 34. ( ) = ( + 2) e ( ) = (1) + ( + 2) = + ( + 2) = ( + 3) ( ) = 0 for = -3; ( ) < 0 for < -3 and ( ) > 0 for > -3. Summary: is decreasing on the interval (- ! , -3); increasing on the interval (-3, ! ). has a local minimum at = -3. 36. ( ) = 4/3 – 7 1/3 ( ) = 4 3 1/3 - 7 3 -2/3 = 4 3 - 7 3 = 4 " 7 3 ( ) = 0 for = 7 4 = 1.75. ( ) < 0 on the interval "# , 7 4 $ % & ( ) = (- ! , 1.75) and ( ) > 0 on the interval (1.75, ! ). Summary: is decreasing on the interval (- ! , 1.75); increasing on the interval (1.75, ! ). has a local minimum at = 1.75. 38. Let ( ) = 4 + 2 - 9 . Since ( ) is a polynomial its critical values (if exist) are the zeros of '( ). '( ) = 4 3 + 2 - 9 Using a graphing utility to approximate the zeros of '( ) we have: '( ) = 0 for " 1.18 Sign chart for ': Increasing Decreasing '( ) ( ) - - - + + + 0 2 1.18 Test Numbers '( ) 0 ! 9( ! ) 2 27( + ) Therefore, is decreasing on (- ! , 1.18); increasing on (1.18, ! ); local minimum at = 1.18. Critical value: = 1.18.
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EXERCISE 5-1 139 40.
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This note was uploaded on 04/21/2009 for the course MATH 121 taught by Professor Hamidy during the Spring '09 term at Miramar College.

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5even - 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 2. (b, c);...

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