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The set of all points, where the functi...

The set of all points, where the function ` f (x) = x/(1+|x|) ` is differentiable, is

A

`(-infty, infty)`

B

`[0, infty)`

C

`(-infty, 0) cup (0, infty)`

D

`(0, infty)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, `f(x) = x/(1+|x|) = {{:(x/(1+x)","x ge 0),(x/(1-x)", "x lt 0):}`
`:." " f'(x) = {{:(((1+x)*1-x*1)/(1+x)^(2)", "x ge 0),(((1-x)*1-x(-1))/(1-x)^(2)", "xlt0):}`
`rArr" "f'(x)={{:(1/((1+x)^(2))", "x ge0),(1/(1-x)^(2)", "xlt0):}`
`:." ""RHD at " x = 0 rArr underset ( x to 0) 1/((1+x)^(2) = 1`
and LHD at ` x = 0 rArr underset( x to 0) lim 1/((1-x)^(2)) = 1`
Hence, f(x) is differentiable for all x.
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