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Question

Let S be the set of all points where the function f(x)=|xπ|(e|x|1)sin|x| is not differentiable, then S is

A
{0}
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B
{π}
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C
{0,π}
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D
ϕ
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Solution

The correct option is D ϕ
f(x)=|xπ|(e|x|1)sin|x|
critical points x=0,π

At x=0
for LHD
f(x)=(xπ)(ex1)sin(x) =(xπ)(ex1)sin(x)
f(x)=(xπ)(ex1)cos(x)+(xπ)(ex)sin(x)+(1)(ex1)sin(x)
f(0)=0
for RHD
f(x)=(πx)(ex1)sin(x)
f(x)=(πx)(ex1)cos(x)+(πx)(ex)sin(x)+(1)(ex1)sin(x)
f(0)=0

At x=π
for LHD
f(x)=(πx)(ex1)sin(x)
f(x)=(πx)(ex1)cos(x)+(πx)(ex)sin(x)+(1)(ex1)sin(x)
f(π)=0
for RHD
f(x)=(xπ)(ex1)sin(x)
f(x)=(xπ)(ex1)cos(x)+(xπ)(ex)sin(x)+(1)(ex1)sin(x)
f(π)=0
By using the first principle of derivative we can say that f(x) is differentiable at x=0,π
f(x) is differentiable for all x ϵ R

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