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Mathematics
Mathematics
Problem
Find a function
g
:
R
→
R
g: \mathbb{R} \to \mathbb{R}
g
:
R
→
R
, continuous in
[
0
,
∞
)
[0, \infty)
[
0
,
∞
)
and positive in
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
satisfying
g
(
0
)
=
1
g(0) = 1
g
(
0
)
=
1
and
1
2
∫
0
x
g
2
(
t
)
d
t
=
1
x
(
∫
0
x
g
(
t
)
d
t
)
2
.
\frac{1}{2} \int_{0}^{x} g^2(t) \, dt = \frac{1}{x} \left( \int_{0}^{x} g(t) \, dt \right)^2.
2
1
∫
0
x
g
2
(
t
)
d
t
=
x
1
(
∫
0
x
g
(
t
)
d
t
)
2
.
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