Thomas Calculus 13th [Solutions]

258 Chapter 4 Applications of Derivatives
26. When 4
2
y x tan x , then y 4 sec x and
3
3
2
y 2sec x tan x . The curve is increasing on
,
6 6
, and decreasing on ,
2 6
and ,
6 2
.
At x there is a local minimum, at x there is
6
6
a local maximum, there are no absolute maxima or
absolute minima. The curve is concave up on
2 , 0 , and is concave down on 0,
2
. At x 0
there is a point of inflection.
2 2
27. When y sin x cos x , then y sin x cos x
cos 2x and y 2sin 2 x . The curve is increasing
on 0, and 3 4 4 , , and decreasing on , 3 . At
4 4
x 0 there is a local minimum, at x there is
4
a local and absolute maximum, at x 3 there is a
4
local and absolute minimum, and at x there is
a local maximum. The curve is concave down on
0, , and is concave up on
2
2 , . At x there is
2
a point of inflection.
28. When y cos x 3 sin x , then y sin x 3 cos x
and y cos x 3 sin x . The curve is increasing on
0, and 4 , 2 , and decreasing on , 4 . At
3 3
3 3
x 0 there is a local minimum, at x there is
3
a local and absolute maximum, at x 4 there is a
3
local and absolute minimum, and at x 2 there is
a local maximum. The curve is concave down on
0, 5 and 11 , 2 , and is concave up on
6 6
5 , 11 . At x 5 and x 11 there are points
6 6
6
6
of inflection.
1/5
29. When y x , then 1 4/5
9/5
y x and y 4 x .
5
25
The curve rises on ( , ) and there are no extrema.
The curve is concave up on ( , 0) and concave
down on (0, ). At x 0 there is a point of inflection.
2/5
30. When y x , then 2 3/5
8/5
y x and y 6 x .
5
25
The curve is rising on (0, ) and falling on ( , 0).
At x 0 there is a local and absolute minimum.
There is no local or absolute maximum. The curve is
concave down on ( , 0) and (0, ). There are no
points of inflection, but a cusp exists at x 0.
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