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Putnam
2006 Putnam
B5
Putnam 2006 B5
Putnam 2006 B5
Source:
December 4, 2006
Putnam
function
integration
calculus
quadratics
inequalities
algebra
Problem Statement
For each continuous function
f
:
[
0
,
1
]
→
R
,
f: [0,1]\to\mathbb{R},
f
:
[
0
,
1
]
→
R
,
let
I
(
f
)
=
∫
0
1
x
2
f
(
x
)
d
x
I(f)=\int_{0}^{1}x^{2}f(x)\,dx
I
(
f
)
=
∫
0
1
x
2
f
(
x
)
d
x
and
J
(
f
)
=
∫
0
1
x
(
f
(
x
)
)
2
d
x
.
J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.
J
(
f
)
=
∫
0
1
x
(
f
(
x
)
)
2
d
x
.
Find the maximum value of
I
(
f
)
−
J
(
f
)
I(f)-J(f)
I
(
f
)
−
J
(
f
)
over all such functions
f
.
f.
f
.
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