The set of points where the function `f ` given by `f(x) - |2x-1|` sinx is differentiable is

Options

A. `R`
B. `R - (1/2)`
C. `(0,oo)`
D. None of these

Answer

B

Explanation

We have, ` f(x)=|2x-1|sinx`
At `x = 1/2,f(x)` is not differentiable.
Hence , `f(x)` is differentiable in `R = (1/2)`
`:' Rf'(1/2)= underset(hrarr0)(lim)(f(1/2+h)-f(1/2))/(h)`
`=underset(hrarr0)(lim)(|2(1/2+h)-1|sin(1/2+h)-0)/(h)`
`=underset(hrarr0)(lim)(|2h|.sin((1+2h)/(2)))/(h)=2.sin'1/2`
and ` Lf'(1/2)=underset(hrarr0)(lim)(f(1/2-h)-f(1/2))/(-h)`
`=underset(hrarr0)(lim)(|2(1/2-h)^(-1)|-sin(1/2-h)-0)/(-h)`
`=underset(hrarr0)lim(|0-2h|-sin(1/2-h))/(-h)=-2sin(1/2)`
`:'Rf'(1/2)neLf'(1/2)`
So, `f(x)` is not differentiable at `x = 1/2`.