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JEE Advanced 2015 Paper 2 Mathematics Q20
Mathematics
Problem
PARAGRAPH 2 Let
F
:
R
→
R
F:\mathbb R\to\mathbb R
F
:
R
→
R
be a thrice differentiable function. Suppose that
F
(
1
)
=
0
F(1)=0
F
(
1
)
=
0
,
F
(
3
)
=
−
4
F(3)=-4
F
(
3
)
=
−
4
and
F
′
(
x
)
<
0
F'(x)<0
F
′
(
x
)
<
0
for all
x
∈
(
1
/
2
,
3
)
x\in(1/2,3)
x
∈
(
1/2
,
3
)
. Let
f
(
x
)
=
x
F
(
x
)
f(x)=xF(x)
f
(
x
)
=
x
F
(
x
)
for all
x
∈
R
x\in\mathbb R
x
∈
R
. If
∫
1
3
x
2
F
′
(
x
)
d
x
=
−
12
\displaystyle\int_1^3x^2F'(x)\,dx=-12
∫
1
3
x
2
F
′
(
x
)
d
x
=
−
12
and
∫
1
3
x
3
F
′
′
(
x
)
d
x
=
40
\displaystyle\int_1^3x^3F''(x)\,dx=40
∫
1
3
x
3
F
′′
(
x
)
d
x
=
40
, then the correct expression(s) is(are)
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