Thomas Calculus 13th [Solutions]

Section 4.1 Extreme Values of Functions 225
50.
2
2
f ( x ) x f
x 2
( x ) ( x 2 )2 x x (1) 2
2
x 4x
2
2
x 4x
0 x 4x 0 x 0 or x 4; x 4 x undefined
2
2
2
2
( x 2) ( x 2) ( x 2)
( x 2)
2
( x 2) 0 x 2. The domain of the function is ( , 2) (2, ), thus x 2 is not the domain, so the only
critical points are x 0 and x 4
51.
3/2
3/2
3/2
2
y x 32 x y 2x
16 2x
16 2x
16 0 2x
16 0
x x
x
x 0 x 0. The critical points are x 4 and x 0.
x
3/2
4; 2x
16 undefined
x
52.
2
g( x) 2 x x g ( x)
1 x 1 x 0 1 x 0 x 1; 1 x undefined
2
2
2
2x x 2x x
2x x
2
2x x 0 x 0 or x 2. The critical points are x 0, x 1, and x 2.
2
2x
x 0
53. Minimum value is 1 at x 2.
2
54. To find the exact values, note that y 3x
2,
which is zero when x 2
3 . Local maximum at
2 4 6
, 4
3 9
( 0.816, 5.089); local minimum
4 6
at 2 , 4
3 9
(0.816, 2.911)
2
55. To find the exact values, note that y 3x 2x
8
(3x 4)( x 2), which is zero when x 2 or x 4
3 .
Local maximum at ( 2, 17); local minimum at
4 , 41
3 27
2
56. Note that y 5 x ( x 5)( x 3), which is zero at
x 0, x 3, and x 5. Local maximum at (3, 108);
local minimum at (5, 0); (0, 0) is neither a maximum
nor a minimum.
Copyright
2014 Pearson Education, Inc.