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Mathematics
Problem
Q.1 Let
f
:
[
1
,
∞
)
→
R
f:[1,\infty)\rightarrow\mathbb{R}
f
:
[
1
,
∞
)
→
R
be a differentiable function such that
f
(
1
)
=
1
3
f(1)=\frac{1}{3}
f
(
1
)
=
3
1
and
3
∫
1
x
f
(
t
)
d
t
=
x
f
(
x
)
−
x
3
3
,
x
∈
[
1
,
∞
)
3\int_{1}^{x}f(t)dt = xf(x) - \frac{x^3}{3}, x \in [1, \infty)
3
∫
1
x
f
(
t
)
d
t
=
x
f
(
x
)
−
3
x
3
,
x
∈
[
1
,
∞
)
. Let
e
e
e
denote the base of the natural logarithm. Then the value of
f
(
e
)
f(e)
f
(
e
)
is (A)
e
2
+
4
3
\frac{e^2+4}{3}
3
e
2
+
4
(B)
log
e
4
+
e
3
\frac{\log_e 4 + e}{3}
3
l
o
g
e
4
+
e
(C)
4
e
2
3
\frac{4e^2}{3}
3
4
e
2
(D)
e
2
−
4
3
\frac{e^2-4}{3}
3
e
2
−
4
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