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Let f(x)=x|sin x|,x in R. Then...

Let `f(x)=x|sin x|,x in R.` Then

A

f is differentiable for all x, except at `x=etapi,eta=1,2,3,....`

B

f is differentiable for all x, except at `x=etapi,eta=pm1,pm2,pm3,…`

C

f is differentiable for all x, except at `x=etapi,eta=0,1,2,3,….`

D

f is differentiable for all x, except at `x=etapi,eta= 0,pm1,pm,2,pm3,….`

Text Solution

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The correct Answer is:
B
`f(x)=x|sinx|={{:(x sin x"," sin ge i.e.","in[2 npi,2npi2pi]),(-x sin x"," sinx lt0 i in(2 npi+pi,24n pi)):}}`
`f'(npi)=underset(xto 4 pi)(lim)(f(x)-f(npi))/(x-npi)=underset(xto4 pi)(lim)(x|sinx|)/(x-4pi)`
`implies ` dist at x=0
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