Thomas Calculus 13th [Solutions]

264 Chapter 4 Applications of Derivatives
x x
53. y e 2e 3x
x 2 x x x
y
x x ( e ) 3e 2 ( e 2)( e 1)
e 2e
3
x
x
e
e
y | | the graph is increasing
0 ln 2
, 0) and (ln 2, ), decreasing on (0, ln 2); a local
on (
maximum is 1 at x = 0 and a local minimum is
x 2
x x ( e ) 2
1 3 ln 2 at x = ln 2; y e 2e
x
e
y | the graph is concave up on
1
2 ln 2
1
2 ln 2, , concave down on , 1 ln 2 point of
2
inflection at 1 ln 2, 3 ln 2 .
2 2
x x x x
54. y xe y e xe (1 x)
e
y | the graph is increasing on ( , 1)
1
1
and decreasing on (1, ); a local maximum is e at
x = 1; y ( x 1) e
x
( 1) e
x
( x 2) e
x
y | the graph is concave up on
2
(2, ), concave down on ( , 2) point of inflection at
2
(2, 2 e ).
55. y = ln(cos x) y sin x tan x
cos x
y .... ) none ( | ) none ( | ) none ( | ) none ( ...
7 5
2
3
0
3
2
5 7
2 2 2 2 2 2 2 2
the graph is increasing on ..., 52 , 2 , 2 , 0 ,
32 , 2 , ..., decreasing on 2 , 3 , 0, , 2 , 5 ;
2 2 2
2
local maxima are 0 at x = 0, 2 , 4 , ...; y sec x 1
2
cos x
the graph is concave down on 5 , 3 ,
2 2
, , 3 , 5 , ...
2 2 2 2
56.
y
ln x
x
y
x
1
ln x
1
x 2 x 2 ln
2 3/2
the graph is increasing on
local maximum is 2 e at x e
2 ;
x
2x
x
y
( |
0 e
2
(0, e ), decreasing on
2
2
( e , ); a
y
3/2 1
3 1/2
x
2 x
3/2 2 5/2
2 x (2 ln x) 2 x
3ln 8
(2 x ) 4x
y ( | the graph is concave up on
0
8/3
e
concave down on
8/3
( e , ),
8/3
8/3
(0, e ) point of inflection is e , 8 .
4/3
3e
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