Thomas Calculus 13th [Solutions]

2 2
40. When y x , then
x
y 2 x 2 2x
2
and
2 2
x x
3
y 2
4 2x
4 . The curve is falling on ( , 0) and
3 3
x x
(0, 1), and rising on (1, ). There is a local minimum at
x 1. There are no absolute maxima or absolute minima.
3
The curve is concave up on , 2 and (0, ), and
concave down on
at x
3 2.
41. When y x
x 2 , then
3 2, 0 . There is a point of inflection
2 2
3
2 x( x 2) ( x 3)(1) ( x 3)( x 1)
y
2
2
( x 2)
( x 2)
2 2
(2x 4)( x 2) ( x 4x 3)2( x 2)
y
2
.
4
3
( 2)
( 2)
and
The curve
x
x
is rising on ( ,1) and (3, ), and falling on (1, 2) and
(2, 3). There is a local maximum at x 1 and a local
minimum at x 3. The curve is concave down on
( , 2) and concave up on (2, ). There are no points
of inflection because x 2 is not in the domain.
3
Section 4.4 Concavity and Curve Sketching 261
3 3
42. When y x 1, then y x and y
2x
.
3 2/3
3 5/3
( x 1)
( x 1)
The curve is rising on ( , 1), ( 1, 0), and (0, ). There
are no local or absolute extrema. The curve is concave up
on ( , 1) and (0, ), and concave down on ( 1, 0).
There are points of inflection at x 1 and x 0.
2
43. When
8
2
x4 ,
8( x 4)
16 x( x 12)
y then y and y
.
2 2
2 3
x
( x 4)
( x 4)
The curve is falling on ( , 2) and (2, ), and is rising
on ( 2, 2). There is a local and absolute minimum at
x 2, and a local and absolute maximum at x 2. The
curve is concave down on , 2 3 and 0, 2 3 , and
concave up on 2 3, 0 and 2 3, . There are points
of inflection at x 2 3, x 0, and x 2 3. y 0 is a
horizontal asymptote.
2
2
2 4
44. When y 5
, then
3
20x
100 x ( x 3)
y and y
.
4
4 2
4 3
x 5 ( x 5)
( x 5)
The curve is rising on ( , 0), and is falling on (0, ).
There is a local and absolute maximum at x 0, and there
is no local or absolute minimum. The curve is concave up
4
on , 3 and 4 3, , and concave down on
4 3, 0
4
and 0, 3 . There are points of inflection at x 4 3 and
x 4 3. There is a horizontal asymptote of y 0.
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