# 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 œœ curve is concave down on ( 3) and concave up on ±_ß (3 ). At x 3 there is a point of inflection. ß_ œ 22. When y x 5 , then y 5 x(4) 5 œ ± ² ± ˆ‰ ˆ xx x ## # # %% \$ w " 5 5 , and y 3 5 5 ± œ ± ± ˆ ˆ ˆ x5 x x 5 x # \$# ww " 5 5 5 (x 4). The curve is rising ²± œ ± ± ˆ x # on ( ) and (10 ), and falling on ( 10). There is a ±_ß# local maximum at x 2 and a local minimum at x 10. The curve is concave down on ( ) and concave up on ±_ß% ( ). At x 4 there is a point of inflection. œ 23. When y x sin x, then y cos x and y sin x. œ² œ " ² ww w 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. 11 1 ß# œ 24. When y x sin x, then y cos x and y sin x. œ " ± œ w 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 œ 25. When y x , then y x and y x . œ ± "Î& w ±%Î& ww ±*Î& " 52 5 4 The curve rises on ( ) and there are no extrema. ±_ß_ The curve is concave up on ( ) and concave down 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 . œœ œ ± \$Î& w ±#Î& ww ±(Î& 36 52 5 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 ± #Î& w ±\$Î& ww ±)Î& 26 5 The curve is rising on (0 ) and falling on ( ). At 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 inflection, but a cusp exists at x 0. œ 28. When y x , then y x and y x . œ ± %Î& w ±"Î& ww ±'Î& 44 5 The curve is rising on (0 ) and falling on ( ). At 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 inflection, but a cusp exists at x 0. œ 29. When y 2x 3x , then y 2 2x and œ± #Î\$ w ±"Î\$ y x . The curve is rising on ( ) and ww ±%Î\$ œ 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 concave up on ( ) and ( ). There are no 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 ± #Î& w ±\$Î& ±\$Î& ˆ‰ and y x . The curve is rising on (0 1) and ww ±)Î& 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 concave down on ( ) and ( ). There are no points of inflection, but a cusp exists at x 0. œ
Section 4.4 Concavity and Curve Sketching 233 31. When y x x x x , then œ± œ ± #Î\$ #Î\$ &Î\$ ## ˆ‰ 55 y x x x (1 x) and w ±"Î\$ #Î\$ ±"Î\$ œ ± 5 33 3 y x x x (1 2x).

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## This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C04_E - Section 4.4 Concavity and Curve Sketching...

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