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Putnam
1973 Putnam
B4
B4
Part of
1973 Putnam
Problems
(1)
Putnam 1973 B4
Source: Putnam 1973
5/25/2022
(a) On
[
0
,
1
]
[0, 1]
[
0
,
1
]
, let
f
(
x
)
f(x)
f
(
x
)
have a continuous derivative satisfying
0
<
f
′
(
x
)
≤
1
0 <f'(x) \leq1
0
<
f
′
(
x
)
≤
1
. Also suppose that
f
(
0
)
=
0.
f(0) = 0.
f
(
0
)
=
0.
Prove that
(
∫
0
1
f
(
x
)
d
x
)
2
≥
∫
0
1
f
(
x
)
3
d
x
.
\left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.
(
∫
0
1
f
(
x
)
d
x
)
2
≥
∫
0
1
f
(
x
)
3
d
x
.
(b) Show an example in which equality occurs.
Putnam
inequalities
integration
Integral