MathDB
Problems
Contests
Undergraduate contests
ICMC
ICMC 2
5
5
Part of
ICMC 2
Problems
(1)
ICMC 2018/19 Round 1, Problem 5
Source: Imperial College Mathematics Competition 2018/19 - Round 1
8/7/2020
For continuously differentiable function
f
:
[
0
,
1
]
→
R
f : [0, 1] \to\mathbb{R}
f
:
[
0
,
1
]
→
R
with
f
(
1
/
2
)
=
0
f (1/2) = 0
f
(
1/2
)
=
0
, show that
(
∫
0
1
f
(
x
)
d
x
)
2
≤
1
4
∫
0
1
(
f
′
(
x
)
)
2
d
x
\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x
(
∫
0
1
f
(
x
)
d
x
)
2
≤
4
1
∫
0
1
(
f
′
(
x
)
)
2
d
x
college contests