Analysis 1: Exercises 11
1.
Let
f
be continuous on [
a, b
]. Suppose that
f
(
x
) = 0 for all rational
x
∈
[
a, b
]. Prove
that
f
(
x
) = 0 for all
x
∈
[
a, b
].
2.
Prove that the following equations have solutions in [

2
,
0].
(a)
x
3

x
+ 3 = 0;
(b)
x
5
+
x
+ 1 = 0.
3.
Let
f
and
g
be continuous on [
a, b
]. Suppose that
f
(
a
)
< g
(
a
) and
f
(
b
)
> g
(
b
). Prove
that there exists
x
∈
(
a, b
) such that
f
(
x
) =
g
(
x
).
4.
Prove that the equation
x
4

3
x

9 = 0 has at least two solutions in [

2
,
2].
5.
Let
f
: [0
,
1]
→
[0
,
1] be continuous. Prove that
(
∃
x
∈
[0
,
1])
(
f
(
x
) =
x
)
.
6.
Let
I
stand for the
closed
interval
I
:= [
a, b
]. Let
f
:
I
→
R
and
g
:
I
→
R
be
continuous functions on
I
. Define
E
:=
{
x
∈
I
:
f
(
x
) =
g
(
x
)
}
.
Suppose that (
x
n
)
⊆
E
has the property that
x
n
→
α
for some
α
∈
R
. Show that
(a)
α
∈
I
;
(b)
α
∈
E
.
7.
Let
p
(
x
) :=
a
0
x
0
+
a
1
x
1
+
a
2
x
2
+
a
3
x
3
be a cubic polynomial with real coefficients
a
0
, a
1
, a
2
, a
3
∈
R
. Let us assume that
a
3
>
0. Show that
(i) lim
x
→
+
∞
p
(
x
) = +
∞
;
(ii) lim
x
→∞
p
(
x
) =
∞
;
(iii)
p
has at least one real root.
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 Winter '10
 Spyros
 Calculus, Topology, Equations, Continuous function, Metric space, Uniform continuity

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