Thomas Calculus 13th [Solutions]

Section 4.4 Concavity and Curve Sketching 265
x x x x
x
57. 1
( e 1) e e e x
y
e y
e
x x x 2 x 2
1 e e 1 ( e 1) ( e 1)
y
y the graph is increasing on ( , );
x 2 x x x x x x
( e 1) e e 2( e 1) e e (1 e )
x
4 x 3
( e 1) ( e 1)
y | the graph is concave up on
0
( , 0), concave down on (0, ) point of inflection is 0, 1 .
2
x x x x
x ( e 1) e e e x
58. y e y
e
x x 2 x 2
1 e (1 e ) (1 e )
59.
60.
61.
y
y the graph is increasing on ( , );
x 2 x x x x x x
( e 1) e e 2( e 1) e e (1 e )
x 4 x 3
(1 e ) (1 e )
y | the graph is concave up on
0
( , 0), concave down on (0, ) point of inflection is 0, 1 .
2
2
y 2 x x (1 x)(2 x),
y | |
1 2
rising on ( 1, 2), falling on ( , 1) and (2, )
there is a local maximum at x 2 and a local
minimum at x 1; y 1 2 x, y |
concave up on 1
2
a point of inflection at 1
2
1/2
, , concave down on 1 2 ,
x
y
2
x x 6 ( x 3)( x 2), y | |
2 3
rising on ( , 2) and (3, ), falling on ( 2, 3)
there is a local maximum at x 2 and a local minimum at
x 3; y 2x 1, y |
1/2
concave up on 1 2 , , concave down on , 1
2
a point of inflection at x 1
2
y x( x
2
3) , y | | rising on (0, ), falling
0 3
on ( , 0) no local maximum, but there is a local minimum at
x 0; y ( x
2
3) x (2) ( x 3) 3( x 3)( x 1), y
| | concave up on ( ,1) and (3, ), concave
1 3
down on (1, 3) points of inflection at x 1 and x 3
62.
y
2
x (2 x), y | | rising on ( , 2), falling
0 2
on (2, ) there is a local maximum at x 2, but no local
minimum; y 2 x(2 x) 2
x ( 1) x(4 3 x),
y
| | concave up on 0, 4
3
0 4/3
, concave down on , 0
and 4 , points of inflection at 0
3 x 4
3
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