Thomas Calculus 13th [Solutions]

Section 4.3 Monotonic Functions and the First Derivative Test 247
2
54. (a) f ( x) x 2x 3,3 x f ( x ) 2x
2
2
x 2x
3
only critical point in 3 x is at x 3
f [ , f (3) 0 local minimum of 0 at x 3, no local maximum
3
(b) absolute minimum of 0 at x 3, no absolute maximum
(c)
55. (a) g( x) x 2 ,0 x 1
2
x 1
g
[ | ),
0 0.268 1
x
maximum at 0.
(b) absolute minimum of
(c)
2
g ( x ) x 4x
1 only critical point in 0 x 1 is x 2 3 0.268
2 2
( x 1)
3
g 2 3 1.866 local minimum of
4 3 6
3
4 3 6
at x 2 3, no absolute maximum
3
at x 2 3, local
4 3 6
2
56. (a) g ( x ) x , 2 x 1 g ( x ) 8 x only critical point in 2 x 1is x 0
2
2 2
4 x
(4 x )
g ( | ], g (0) 0 local minimum of 0 at x 0, local maximum of 1 3
2 0 1
(b) absolute minimum of 0 at x 0, no absolute maximum
(c)
at x 1.
57. (a) f ( x) sin 2 x , 0 x f ( x) 2cos 2 x, f ( x ) 0 cos 2x 0 critical points are x and x 3
4 4
f [ | | ] , f (0) 0, f 1, f 3 1, f ( ) 0 local maxima are 1 at x
0
3
4 4
4
4 4
and 0 at x , and local minima are 1 at x 3 and 0 at x 0.
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 3 , where the values
4
of f change from negative to positive. The function f has a local
maximum value at x and x
4 , where the values of f change
from positive to negative.
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