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ISM_T11_C04_E - Section 4.4 Concavity and Curve Sketching...

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Section 4.4 Concavity and Curve Sketching 231 21. When y x 5x , then y 5x 20x 5x (x 4) and œ œ œ & % w % $ $ y 20x 60x 20x (x 3). The curve rises on ww $ # # œ œ ( ) and ( ), and falls on ( ). There is a local _ß ! %ß _ !ß % maximum at x 0, and a local minimum at x 4. The œ œ curve is concave down on ( 3) and concave up on (3 ). At x 3 there is a point of inflection. ß _ œ 22. When y x 5 , then y 5 x(4) 5 œ œ ˆ ˆ ˆ ‰ ˆ ‰ x x x # # # # % % $ w " 5 5 , and y 3 5 5 œ œ ˆ ‰ ˆ ˆ ‰ ˆ ‰ ˆ x 5x x 5x # # # # # $ # ww " 5 5 5 (x 4). The curve is rising œ ˆ ‰ ˆ ‰ ˆ x 5 x # # # $ # on ( ) and (10 ), and falling on ( 10). There is a _ß # ß _ local maximum at x 2 and a local minimum at x 10. œ œ The curve is concave down on ( ) and concave up on _ß % ( ). At x 4 there is a point of inflection. %ß _ œ 23. When y x sin x, then y cos x and y sin x. œ œ " œ w ww The curve rises on ( 2 ). At x 0 there is a local and œ 1 absolute minimum and at x 2 there is a local and absolute œ 1 maximum. The curve is concave down on ( ) and concave 1 up on ( ). At x there is a point of inflection. 1 1 1 ß # œ 24. When y x sin x, then y cos x and y sin x. œ œ " œ w ww The curve rises on ( 2 ). At x 0 there is a local and œ 1 absolute minimum and at x 2 there is a local and absolute œ 1 maximum. The curve is concave up on ( ) and concave 1 down on ( ). At x there is a point of inflection. 1 1 1 ß # œ 25. When y x , then y x and y x . œ œ œ "Î& w %Î& ww *Î& " 5 25 4 The curve rises on ( ) and there are no extrema. _ß _ The curve is concave up on ( ) and concave down _ß ! on ( ). At x 0 there is a point of inflection. !ß _ œ
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232 Chapter 4 Applications of Derivatives 26. When y x , then y x and y x . œ œ œ $Î& w #Î& ww (Î& 3 6 5 25 The curve rises on ( ) and there are no extrema. _ß _ The curve is concave up on ( ) and concave down _ß ! on ( ). At x 0 there is a point of inflection. !ß _ œ 27. When y x , then y x and y x . œ œ œ #Î& w $Î& ww )Î& 2 6 5 25 The curve is rising on (0 ) and falling on ( ). At ß _ _ß ! x 0 there is a local and absolute minimum. There is œ no local or absolute maximum. The curve is concave down on ( ) and ( ). There are no points of _ß ! !ß _ inflection, but a cusp exists at x 0. œ 28. When y x , then y x and y x . œ œ œ %Î& w "Î& ww 'Î& 4 4 5 25 The curve is rising on (0 ) and falling on ( ). At ß _ _ß ! x 0 there is a local and absolute minimum. There is œ no local or absolute maximum. The curve is concave down on ( ) and ( ). There are no points of _ß ! !ß _ inflection, but a cusp exists at x 0. œ 29. When y 2x 3x , then y 2 2x and œ œ #Î$ w "Î$ y x . The curve is rising on ( ) and ww %Î$ œ _ß ! 2 3 ( ), and falling on ( ). There is a local maximum "ß _ !ß " at x 0 and a local minimum at x 1. The curve is œ œ concave up on ( ) and ( ). There are no _ß ! !ß _ points of inflection, but a cusp exists at x 0. œ 30. When y 5x 2x, then y 2x 2 2 x 1 œ œ œ #Î& w $Î& $Î& ˆ and y x . The curve is rising on (0 1) and ww )Î& œ ß 6 5 falling on ( 0) and ( ). There is a local minimum "ß _ at x 0 and a local maximum at x 1. The curve is œ œ concave down on ( ) and ( ). There are no _ß ! !ß _ points of inflection, but a cusp exists at x 0. œ
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Section 4.4 Concavity and Curve Sketching 233 31. When y x x x x , then œ œ #Î$ #Î$ &Î$ # # ˆ 5 5 y x x x (1 x) and w
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