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Mathematics
Mathematics
Problem
Let
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f: (0, \infty) \to (0, \infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
be a differentiable function satisfying,
x
∫
0
x
(
1
−
t
)
f
(
t
)
d
t
=
∫
0
x
t
f
(
t
)
d
t
x
∈
R
+
and
f
(
1
)
=
1.
Determine
f
(
x
)
.
x \int_{0}^{x} (1-t)f(t)dt = \int_{0}^{x} tf(t)dt \quad x \in R^{+} \text{ and } f(1) = 1. \text{ Determine } f(x).
x
∫
0
x
(
1
−
t
)
f
(
t
)
d
t
=
∫
0
x
t
f
(
t
)
d
t
x
∈
R
+
and
f
(
1
)
=
1.
Determine
f
(
x
)
.
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