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JEE Advanced 2015 Paper 1 Mathematics Q16
Mathematics
Problem
Let
g
:
R
→
R
g:\mathbb R\to\mathbb R
g
:
R
→
R
be a differentiable function with
g
(
0
)
=
0
g(0)=0
g
(
0
)
=
0
,
g
′
(
0
)
=
0
g'(0)=0
g
′
(
0
)
=
0
and
g
′
(
1
)
≠
0
g'(1)\ne0
g
′
(
1
)
=
0
. Let
f
(
x
)
=
{
x
∣
x
∣
g
(
x
)
,
x
≠
0
,
0
,
x
=
0
,
f(x)=\begin{cases}\dfrac{x}{|x|}g(x),&x\ne0,\\0,&x=0,\end{cases}
f
(
x
)
=
⎩
⎨
⎧
∣
x
∣
x
g
(
x
)
,
0
,
x
=
0
,
x
=
0
,
and
h
(
x
)
=
e
∣
x
∣
h(x)=e^{|x|}
h
(
x
)
=
e
∣
x
∣
for all
x
∈
R
x\in\mathbb R
x
∈
R
. Let
(
f
∘
h
)
(
x
)
(f\circ h)(x)
(
f
∘
h
)
(
x
)
denote
f
(
h
(
x
)
)
f(h(x))
f
(
h
(
x
))
and
(
h
∘
f
)
(
x
)
(h\circ f)(x)
(
h
∘
f
)
(
x
)
denote
h
(
f
(
x
)
)
h(f(x))
h
(
f
(
x
))
. Then which of the following is (are) true?
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