Thomas Calculus 13th [Solutions]

Section 4.4 Concavity and Curve Sketching 263
49. When
1/ x
y xe , then
1/ x
1/ x 1/ x
y xe e e 1 1
2
x
x
1/ x 1/ x 1/ x
1/ x
and y e 1 1 1 e e 1 e .
2 x 2 2 x 3
x x x x
The curve is rising on (1, ) and ( , 0) and falling on
(0, 1). The curve is concave down on ( , 0) and
concave up on (0, ). There is a local minimum of e at
x = 1, but there are no inflection points.
50. y
x x x
e ( x 1) e
y xe e
x
x 2 2
x x
y | | the graph is rising on
0 1
(1, ), falling on ( , 0) and (0, 1); a local minimum is e
2 x x x x x 2
x
x ( xe e e ) ( xe e )(2 x) ( x 2x 2) e
at x = 1; y
4 3
x
x
y | the graph is concave up on
0
(0, ), concave down on ( , 0), but has no inflection
points.
51.
2
y ln(3 x ) y 2x 2x
2 2
3 x x 3
y ( | ) the graph is rising on
3 0 3
3, 0 , falling on 0, 3 ; a local minimum is ln 3 at
52.
x = 0;
y
2 2
( x 3)(2) (2 x)(2 x) 2( x 3)
2 2 2 2
( x 3) ( x 3)
y ( ) the graph is concave down on
3 3
3, 3 .
2
y x(ln x)
y x 2ln x 1
2
(1) (ln x) x
ln x(2 ln x)
y ( | | the graph is rising on
0
2
e 1
2
(0, e ) and (1, ), falling on
2
( e , 1); a local
2
maximum is 4e at x
2
e and a local minimum is 0
2(1 ln )
at x = 1; ln 1 1
x
y x (2 ln x)
x x x
y ( | the graph is concave up on
0
1
e
1
( e , ), concave down on
1 1
inflection at ( e , e ).
1
(0, e ) point of
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