# ISM_T11_C04_F - Section 4.4 Concavity and Curve Sketching...

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Section 4.4 Concavity and Curve Sketching 239 58. y (x 2) , y )( rising on (2 ), w ±"Î\$ w œ ± œ ±±± ²²² Ê ß_ # falling on ( ) no local maximum, but a local ±_ß# Ê minimum at x 2; y (x 2) , œœ ± ± ww ±%Î\$ 1 3 y concave down on ( 2) and ww œ ±±± ±±± Ê ±_ß # ( ) no points of inflection, but there is a cusp at #ß_ Ê x2 œ 59. y x (x 1), y rising on w ±#Î\$ w œ ± œ ±±± ±±± ³ ²²² Ê ! " ( ), falling on ( ) no local maximum, but a "ß_ ±_ß" Ê local minimum at x 1; y x x ² ww ±#Î\$ ±&Î\$ " 33 2 x( x 2 ) , y ) ( œ ² œ ²²² ³ ±±± ²²² ±# ! " ±&Î\$ ww 3 concave up on ( 2) and ( ), concave down on Ê± _ ß ± ! ß _ ( ) points of inflection at x 2 and x 0, and a ±#ß! Ê œ ± œ vertical tangent at x 0 œ 60. y x (x 1), y rising on w ±%Î& w œ ² œ ±±± ³ ²²² ²²² Ê ±" ! ( 0) and ( ), falling on ( ) no local ±"ß !ß_ ±_ß±" Ê maximum, but a local minimum at x 1; œ± y x x x (x 4), ww ±%Î& ±*Î& ±*Î& "" œ±œ ± 55 5 4 y concave up on ( 0) and ww œ ²²² ±±± ³ ²²² Ê ±_ß ! % (4 ), concave down on (0 4) points of inflection at ß Ê x 0 and x 4, and a vertical tangent at x 0 œ 61. y , y rising on x, x 0 2x, x 0 ww œ œ ²²² ³ ²²² Ê ±# Ÿ ´ ! o ( ) no local extrema; y , 2, x 0 2, x 0 ±_ß_ Ê œ ±µ ´ ww o y concave up on ( ), concave ww œ ±±± ²²² Ê ! down on ( ) a point of inflection at x 0 ±_ß! Ê œ 62. y , y rising on x, x 0 x , x 0 # # œ œ ±±± ³ ²²² Ê ±Ÿ ´ ! o ( ), falling on ( ) no local maximum, but a local minimum at x 0; y , 2x, x 0 2x, x 0 ´ ww o y concave up on ( ) ww œ ²²² ³ ²²² Ê ±_ß_ ! no point of inflection Ê

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240 Chapter 4 Applications of Derivatives 63. The graph of y f (x) the graph of y f(x) is concave œÊ œ ww up on ( ), concave down on ( ) a point of !ß_ ±_ß! Ê inflection at x 0; the graph of y f (x) œœ w y the graph y f(x) has Ê œ ²²² ³ ±±± ³ ²²² Ê œ w both a local maximum and a local minimum 64. The graph of y f (x) y the œ Ê œ ²²² ³ ±±± Ê ww ww graph of y f(x) has a point of inflection, the graph of œ y f (x) y the graph of œ Ê œ ±±± ³ ²²² ³ ±±± Ê ww y f(x) has both a local maximum and a local minimum œ 65. The graph of y f (x) y œ Ê œ ±±± ³ ²²² ³ ±±± ww ww the graph of y f(x) has two points of inflection, the Êœ graph of y f (x) y the graph of œ Ê œ ±±± ³ ²²² Ê y f(x) has a local minimum œ 66. The graph of y f (x) y the œ Ê œ ²²² ³ ±±± Ê ww ww graph of y f(x) has a point of inflection; the graph of œ y f (x) y the graph of œ Ê œ ±±± ³ ²²² ³ ±±± Ê y f(x) has both a local maximum and a local minimum œ 67. Point y P Q R S T w ±² ²! ²± ±±
Section 4.4 Concavity and Curve Sketching 241 68. 69. 70. 71. Graphs printed in color can shift during a press run, so your values may differ somewhat from those given here. (a) The body is moving away from the origin when displacement is increasing as t increases, 0 t 2 and kk ±± 6 t 9.5; the body is moving toward the origin when displacement is decreasing as t increases, 2 t 6 and 9.5 t 15 (b) The velocity will be zero when the slope of the tangent line for y s(t) is horizontal. The velocity is zero œ when t is approximately 2, 6, or 9.5 sec.

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ISM_T11_C04_F - Section 4.4 Concavity and Curve Sketching...

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