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Moldova National Olympiad
2001 Moldova National Olympiad
Problem 7
integral inequality
integral inequality
Source: 2001 Moldova MO Grade 12 P7
April 13, 2021
calculus
integration
inequalities
Problem Statement
Let
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
be a continuously differentiable function such that
f
(
x
0
)
=
0
f(x_0)=0
f
(
x
0
)
=
0
for some
x
0
∈
[
0
,
1
]
x_0\in[0,1]
x
0
∈
[
0
,
1
]
. Prove that
∫
0
1
f
(
x
)
2
d
x
≤
4
∫
0
1
f
’
(
x
)
2
d
x
.
\int^1_0f(x)^2dx\le4\int^1_0f’(x)^2dx.
∫
0
1
f
(
x
)
2
d
x
≤
4
∫
0
1
f
’
(
x
)
2
d
x
.
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