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SEEMOUS
2013 SEEMOUS
Problem 3
Problem 3
Part of
2013 SEEMOUS
Problems
(1)
integral inequality, condition 1≥int|f'(x)|^2 from 0 to 1
Source: SEEMOUS 2013 P3
6/8/2021
Find the maximum value of
∫
0
1
∣
f
′
(
x
)
∣
2
∣
f
(
x
)
∣
1
x
d
x
\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx
∫
0
1
∣
f
′
(
x
)
∣
2
∣
f
(
x
)
∣
x
1
d
x
over all continuously differentiable functions
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
with
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
∫
0
1
∣
f
′
(
x
)
∣
2
d
x
≤
1.
\int^1_0|f'(x)|^2dx\le1.
∫
0
1
∣
f
′
(
x
)
∣
2
d
x
≤
1.
calculus
integration
inequalities