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MA1a HW 5 Solutions
November 7, 2007
Problem 1
(p191: 9,10,11)
In the following exercises, (a) ﬁnd all points
x
such that
f
0
(
x
) = 0; (b)
examine the sign of
f
0
and determine those intervals in which
f
is monotonic; (c)
examine the sign of
f
00
and determine those intervals in which
f
0
is monotonic;
(d) make a sketch of the graph of
f
. In each case, the function is deﬁned for all
x
for which the given formula for
f
(
x
) is meaningful.
9.
f
(
x
) =
x/
(1 +
x
2
)
.
10.
f
(
x
) = (
x
2

4)
/
(
x
2

9)
.
11.
f
(
x
) = sin
2
x.
Solution 1.
9. We compute
f
0
(
x
) = (1

x
2
)
/
(1+
x
2
)
2
and
f
00
(
x
) = 2
x
(
x
2

3)
/
(1+
x
2
)
3
.
Then
f
0
(
x
) = 0 precisely when
x
=
±
1, and
f
0
is continuous, so its sign is
constant in the intervals (
∞
,

1)
,
(

1
,
1) and (1
,
∞
). In these intervals its sign
is (

)
,
(+) and (

), respectively, and
f
is monotonic in the intervals (
∞
,

1]
,
[

1
,
1] and [1
,
∞
), because these are the intervals in which either
f
0
(
x
)
≥
0 for
f
0
(
x
)
≤
0 for all
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This note was uploaded on 02/22/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.
 Fall '08
 Borodin,A
 Calculus, Linear Algebra, Algebra

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