Thomas Calculus 13th [Solutions]

248 Chapter 4 Applications of Derivatives
58. (a) f ( x) sin x cos x, 0 x 2 f ( x) cos x sin x, f ( x) 0 tan x 1 critical points are x 3
4
and x 7 f [ | | ] , f (0) 1, f 3 2, f 7 2, f (2 ) 1 local
4
0
3 7
4 4
2
4 4
maxima are 2 at x 3 and 1 at x 2 , and local minima are 2 at x 7 and 1 at x 0.
4
4
(b) The graph of f rises when f 0, falls when
f 0, and has local extreme values where
f 0. The function f has a local minimum
value at x 0 and x 74 , where the values
of f change from negative to positive. The
function f has a local maximum value at x 2
and x 34 , where the values of f change from
positive to negative.
59. (a) f ( x) 3 cos x sin x,0 x 2 f ( x) 3 sin x cos x, f ( x) 0 tan x 1 critical points
3
are x and x 7 f [ | | ] , f (0) 3, f 2, f 7 2, f (2 ) 3
6 6
0
7
6 6
2
6 6
local maxima are 2at x and 3 at x 2 , and local minima are 2 at x 7 and 3 at x 0.
6
6
(b) The graph of f rises when f 0, falls when
f 0, and has local extreme values where
f 0. The function f has a local minimum
value at x 0 and x 7 , where the values
6
of f change from negative to positive. The
function f has a local maximum value at
x 2 and x , where the values of f
6
change from positive to negative.
60. (a) f ( x) 2x tan x,
x
and x
4 4
2
x f ( x) 2 sec x , f ( x) 0
2 2
f ( | | ) , f
4 2
2
4
4 2
is 2
1 at x 4
, and local minimum is 1 at x .
2 4
(b) The graph of f rises when f 0, falls when
f 0, and has local extreme values where
f 0. The function f has a local minimum
value at x
4 , where the values of f change
from negative to positive. The function f has a
local maximum value at x
4 , where the
values of f change from positive to negative.
2
sec x 2 critical points are
1,
f 1 local maximum
4 2
61. (a)
f ( x) x 2sin x f ( x) 1 cos x , f ( x ) 0 cos x 1 a critical point at x 2
2 2
2 2
2 2
3
f [ | ] and f (0) 0, f 2 3, f (2 ) local maxima are
3 3
0 2 /3 2
0 at x 0 and at x 2 , a local minimum is 3 at x 2
3 3
Copyright
2014 Pearson Education, Inc.