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2023 IMC
7
7
Part of
2023 IMC
Problems
(1)
Necessary values of f'-f
Source: 2023 IMC P7
8/3/2023
Let
V
V
V
be the set of all continuous functions
f
:
[
0
,
1
]
→
R
f\colon [0,1]\to \mathbb{R}
f
:
[
0
,
1
]
→
R
, differentiable on
(
0
,
1
)
(0,1)
(
0
,
1
)
, with the property that
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
. Determine all
α
∈
R
\alpha \in \mathbb{R}
α
∈
R
such that for every
f
∈
V
f\in V
f
∈
V
, there exists some
ξ
∈
(
0
,
1
)
\xi \in (0,1)
ξ
∈
(
0
,
1
)
such that
f
(
ξ
)
+
α
=
f
′
(
ξ
)
f(\xi)+\alpha = f'(\xi)
f
(
ξ
)
+
α
=
f
′
(
ξ
)
calculus
IMC
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IMC 2023