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JEE Advanced 2014 Paper 2 Mathematics Q14
Mathematics
Problem
Paragraph for Questions 13 and 14 Given that for each
a
∈
(
0
,
1
)
a\in(0,1)
a
∈
(
0
,
1
)
,
lim
h
→
0
+
∫
h
1
−
h
t
−
a
(
1
−
t
)
a
−
1
d
t
\displaystyle \lim_{h\to0^+}\int_h^{1-h}t^{-a}(1-t)^{a-1}\,dt
h
→
0
+
lim
∫
h
1
−
h
t
−
a
(
1
−
t
)
a
−
1
d
t
exists. Let this limit be
g
(
a
)
g(a)
g
(
a
)
. In addition, it is given that the function
g
(
a
)
g(a)
g
(
a
)
is differentiable on
(
0
,
1
)
(0,1)
(
0
,
1
)
. The value of
g
′
(
1
2
)
g'\!\left(\dfrac12\right)
g
′
(
2
1
)
is
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