Thomas Calculus 13th [Solutions]

260 Chapter 4 Applications of Derivatives
2/3
36. When y x ( x 5)
5/3 2/3
x 5 x , then
5 2/3 10 1/3
y x x
5 1/3
x ( x 2) and
3 3 3
10 1/3
y x
10 4/3
9 9 x 10 4/3
x ( x 1). The curve
9
is rising on ( , 0) and (2, ), and falling on (0, 2).
There is a local minimum at x 2 and a local
maximum at x 0. The curve is concave up on
( 1, 0) and (0, ), and concave down on ( , 1).
There is a point of inflection at x 1 and a cusp
at x 0.
2
2 1/2
37. When y x 8 x x(8 x ) , then
2 1/2
2 1/2
y (8 x ) ( x) 1
(8 x ) ( 2 x)
2
2 1/2 2 2(2 x)(2 x)
(8 x ) (8 2 x )
and
2 2 x 2 2 x
y
3
1 2 2
2
(8 x ) ( 2 x )(8 2 x 1
2
) (8 x )
2
( 4 x)
2
2
2 x( x 12)
2 3
(8 x )
. The curve is rising on ( 2, 2), and falling
on 2 2, 2 and 2, 2 2 . There are local minima
x 2 and x 2 2, and local maxima at x 2 2
and x 2. The curve is concave up on 2 2, 0 and
concave down on 0, 2 2 . There is
a point of inflection at x 0.
2 1/2
2 3/2
38. When y (2 x ) , then y 3
2 (2 x ) ( 2 x)
2
3x 2 x 3x 2 x 2 x and
y
2 1/2
2 1/2
( 3)(2 x ) ( 3 x) (2 x ) ( 2 x)
2
6(1 x)(1 x)
. The curve is rising on 2, 0 and
2 x 2 x
falling on 0, 2 . There is a local maximum at x 0,
and local minima at x 2. The curve is concave
down on ( 1, 1) and concave up on 2, 1 and
1, 2 . There are points of inflection at x 1.
1
39. When
2
y 16 x , then
16
2 3/2
(16 )
x
2
16 x
y and
y . The curve is rising on ( 4, 0) and
x
falling on (0, 4). There is a local and absolute
maximum at x 0 and local and absolute minima at
x 4 and x 4. The curve is concave down on
( 4, 4). There are no points of inflection.
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