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Mathematics
Mathematics
Problem
Let
f
:
[
0
,
2
]
→
R
f: [0, 2] \to \mathbb{R}
f
:
[
0
,
2
]
→
R
be a function which is continuous on
[
0
,
2
]
[0, 2]
[
0
,
2
]
and is differentiable on
(
0
,
2
)
(0, 2)
(
0
,
2
)
with
f
(
0
)
=
1
f(0) = 1
f
(
0
)
=
1
. Let
F
(
x
)
=
∫
0
x
2
f
(
t
)
d
t
for
x
∈
[
0
,
2
]
.
F(x) = \int_{0}^{x^2} f(\sqrt{t}) dt \quad \text{for } x \in [0, 2].
F
(
x
)
=
∫
0
x
2
f
(
t
)
d
t
for
x
∈
[
0
,
2
]
.
If
F
′
(
x
)
=
f
′
(
x
)
F'(x) = f'(x)
F
′
(
x
)
=
f
′
(
x
)
for all
x
∈
(
0
,
2
)
x \in (0, 2)
x
∈
(
0
,
2
)
, then
F
(
2
)
F(2)
F
(
2
)
equals [JEE(Advanced) 2014] (A)
e
2
−
1
e^2 - 1
e
2
−
1
(B)
e
4
−
1
e^4 - 1
e
4
−
1
(C)
e
−
1
e - 1
e
−
1
(D)
e
4
e^4
e
4
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