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Math 310 
hw
10 solutions
29.2, 29.14, 29.18;
Monday, 30 Nov 2009
29.2 Prove that

cos
x

cos
y
 ≤ 
x

y

for all
x,y
∈
R
.
Let
f
:
R
→
R
be deﬁned by
f
(
x
) := cos
x
for
x
∈
R
. We will use the facts that
f
is diﬀerentiable on
R
and that
f
0
(
x
) =

sin
x
for all
x
∈
R
. Let
x,y
∈
R
. If
x
=
y
there is nothing to prove, so we may assume that
x
6
=
y
.
Since
f
is diﬀerentiable on
R
we may apply the Mean Value Theorem to conclude that
f
(
x
)

f
(
y
) =
f
0
(
c
)(
x

y
)
for some
c
between
x
and
y
. Since

f
0
(
c
)

=
 
sin
c
 ≤
1. Hence,

cos
x

cos
y

=

f
(
x
)

f
(
y
)

=

f
0
(
c
)
 · 
x

y
 ≤
1
· 
x

y

=

x

y

,
as desired.
±
29.14 Suppose that
f
is diﬀerentiable on
R
, that 1
≤
f
0
(
x
)
≤
2 for
x
∈
R
, and that
f
(0) = 0. Prove that
x
≤
f
(
x
)
≤
2
x
for all
x
≥
0.
Let
x
≥
0. Since
f
(0) = 0, the desired condition holds for
x
= 0; so we may assume that
x >
0. Since
f
is
continuous on [0
,x
] and diﬀerentiable on (0
,x
), we may invoke the Mean Value Theorem to conclude that there
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 Spring '10
 janeday

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