ANALYSIS
EXERCISE 12–SOLUTIONS
1. Since the function
f
is continuous on [
a, b
] there are
{
x
1
, x
2
} ⊂
[
a, b
] such
that
f
(
x
1
) =
m, f
(
x
2
) =
M
. If
m
=
M
then
f
is a constant function,
and the result is vacuously true (as (
m,M
) =
∅
.) Otherwise set
x
0
=
min
{
x
1
, x
2
}
, x
00
= max
{
x
1
, x
2
}
and consider
f
on the interval [
x
0
, x
00
].
The statement now follows from the intermediatevalue theorem.
2.
lim
h
→
0
f
(
a
+
h
) +
g
(
a
+
h
)

f
(
a
)

g
(
a
)
h
= lim
h
→
0
±
f
(
a
+
h
)

f
(
a
)
h
+
g
(
a
+
h
)

g
(
a
)
h
²
=
f
0
(
a
) +
g
0
(
a
)
.
3. Let
n
= 1. Then
f
(
x
) =
x
, and
f
0
(
x
) = 1. Suppose that the statement is
true for
n
=
k
, i.e.
f
k
(
x
) =
x
k
is diﬀerentiable and
f
0
k
(
a
) =
ka
k

1
. Let
n
=
k
+ 1. Then
f
k
+1
=
x
k
+1
=
x
·
f
k
(
x
) is diﬀerentiable as a product of
two diﬀerentiable functions and
f
0
k
+1
(
a
) = 1
·
f
k
(
a
) +
a
·
f
0
k
(
a
) =
a
k
+
a
·
ka
k

1
= (
k
+ 1)
a
k
.
4. Assume that
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 Winter '10
 Spyros
 Calculus, Mean Value Theorem, Rolle, Rolle's theorem

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