STEP II, 2002
3
4
Give a sketch to show that, if f(
x
)
>
0 for
p < x < q
, then
R
q
p
f(
x
) d
x >
0 .
(i)
By considering f(
x
) =
ax
2

bx
+
c
show that, if
a >
0 and
b
2
<
4
ac
, then 3
b <
2
a
+6
c
.
(ii)
By considering f(
x
) =
a
sin
2
x

b
sin
x
+
c
show that, if
a >
0 and
b
2
<
4
ac
, then
4
b <
(
a
+ 2
c
)
π .
(iii)
Show that, if
a >
0,
b
2
<
4
ac
and
q > p >
0 , then
b
ln(
q/p
)
< a
±
1
p

1
q
²
+
c
(
q

p
)
.
5
The numbers
x
n
, where
n
= 0, 1, 2,
. . .
, satisfy
x
n
+1
=
kx
n
(1

x
n
)
.
(i)
Prove that, if 0
< k <
4 and 0
< x
0
<
1, then 0
< x
n
<
1 for all
n
.
(ii)
Given that
x
0
=
x
1
=
x
2
=
···
=
a
, with
a
6
= 0 and
a
6
= 1, ﬁnd
k
in terms of
a
.
(iii)
Given instead that
x
0
=
x
2
=
x
4
=
···
=
a
, with
a
6
= 0 and
a
6
= 1, show that
ab
3

b
2
+ (1

a
) = 0, where
b
=
k
(1

a
) . Given, in addition, that
x
1
6
=
a
, ﬁnd the
possible values of
k
in terms of
a
.