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Vojtěch Jarník IMC
2022 VJIMC
1
integral inequality, existence of function
integral inequality, existence of function
Source: VJIMC 2022 2.1
April 11, 2022
calculus
integration
inequalities
function
Problem Statement
Determine whether there exists a differentiable function
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
such that
f
(
0
)
=
f
(
1
)
=
1
,
∣
f
′
(
x
)
∣
≤
2
for all
x
∈
[
0
,
1
]
and
∣
∫
0
1
f
(
x
)
d
x
∣
≤
1
2
.
f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.
f
(
0
)
=
f
(
1
)
=
1
,
∣
f
′
(
x
)
∣
≤
2
for all
x
∈
[
0
,
1
]
and
∫
0
1
f
(
x
)
d
x
≤
2
1
.
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