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Vojtěch Jarník IMC
2008 VJIMC
Problem 2
integral inequality involving f'(x)^2 and f(x)^(-2)
integral inequality involving f'(x)^2 and f(x)^(-2)
Source: VJIMC 2008 2.2
June 16, 2021
calculus
integration
inequalities
Problem Statement
Find all continuously differentiable functions
f
:
[
0
,
1
]
→
(
0
,
∞
)
f:[0,1]\to(0,\infty)
f
:
[
0
,
1
]
→
(
0
,
∞
)
such that
f
(
1
)
f
(
0
)
=
e
\frac{f(1)}{f(0)}=e
f
(
0
)
f
(
1
)
=
e
and
∫
0
1
d
x
f
(
x
)
2
+
∫
0
1
f
′
(
x
)
2
d
x
≤
2.
\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.
∫
0
1
f
(
x
)
2
d
x
+
∫
0
1
f
′
(
x
)
2
d
x
≤
2.
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