Thomas Calculus 13th [Solutions]

222 Chapter 4 Applications of Derivatives
27.
3 1/3 1 2/3
h( x) x x h ( x)
x a critical point
3
at x 0; h( 1) 1, h (0) 0, h (8) 2 the
absolute maximum is 2 at x 8 and the absolute
minimum is 1 at x 1
28.
2/3 1/3
h( x) 3 x h ( x) 2x a critical point at
x 0; h( 1) 3, h(0) 0, h (1) 3 the absolute
maximum is 0 at x 0 and the absolute minimum is
3 at x 1 and x 1
29.
2 2 1/2
g( x) 4 x (4 x )
2 1/2
g ( x) 1 (4 x ) ( 2 x ) x critical
2
2
4 x
points at x 2 and x 0, but not at x 2 because 2
is not in the domain;
g( 2) 0, g(0) 2, g (1) 3 the absolute
maximum is 2 at x 0 and the absolute minimum is
0 at x 2
30.
2 2 1/2
g( x) 5 x (5 x )
1
2
2 1/2
g ( x) (5 x ) ( 2 x) x
critical points at x 5 and x 0, but not at
x 5 because 5 is not in the domain;
f 5 0, f (0) 5
5
x
2
the absolute maximum is 0 at x 5 and the
absolute minimum is 5 at x 0
31.
32.
f ( ) sin f ( ) cos is a critical
point, but
2 is not a critical point because 2 is
not interior to the domain; f 1, f 1,
5 1
6 2
2
2 2
f the absolute maximum is 1 at
and the absolute minimum is 1 at
2
f ( ) tan f ( ) sec f has no critical
points in , . The extreme values therefore
3 4
occur at the endpoints: f 3 and f 1
the absolute maximum is 1 at
the absolute minimum is
3
3 at
3
2
4 and
4
2
Copyright
2014 Pearson Education, Inc.