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Past exam of the mathematics course of the University of Cambridge
/
2022
/
ia
/
Paper 1
/
11D
/
b
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2022
ia
Paper 1
11D
b
Created
2026-09-24
Updated
2026-09-25
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Let
m
=
b
−
a
f
(
b
)
−
f
(
a
)
,
h
(
x
)
=
f
(
x
)
−
f
(
a
)
−
m
(
x
−
a
)
.
(1)
Then
h
(
a
)
=
h
(
b
)
=
0
. Since
f
is not linear,
h
(
c
)
=
0
for some
c
∈
(
a
,
b
)
. If
h
(
c
)
>
0
, the
mean value theorem
on
[
a
,
c
]
gives
a
point
ξ
with
h
′
(
ξ
)
=
c
−
a
h
(
c
)
−
h
(
a
)
>
0.
(2)
If
h
(
c
)
<
0
, apply it on
[
c
,
b
]
to obtain
h
′
(
ξ
)
=
b
−
c
h
(
b
)
−
h
(
c
)
>
0.
(3)
In either case
h
′
=
f
′
−
m
, and therefore
f
′
(
ξ
)
>
b
−
a
f
(
b
)
−
f
(
a
)
.
(4)
Ancestors
(11)
b
11D
Paper 1
ia
2022
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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