If `f(x)=sin^(-1)(2xsqrt(1-x^(2))), x in [-1,1]`. Then

Options

A. `f'(x)=(2)/(sqrt(1-x^(2))),"for all "x in(-1,1)`
B. `f'(x)={{:(,(2)/(sqrt(1-x^(2))),"If "|x|lt(1)/(sqrt2)),(,(-2)/(sqrt(1-x^(2))),"If "(1)/(sqrt2) lt |x| lt (1)/(2)):}`
C. `f'(x)={{:(,(-2)/(sqrt(1-x^(2))),"If "|x|lt(1)/(sqrt2)),(,(2)/(sqrt(1-x^(2))),"If "(1)/(sqrt2) lt |x| lt 1):}`
D. f(x) exists for all `x in [-1,1]`

Answer

B

Explanation

We have
`f(x)=sin^(-1) (2x sqrt(1-x^(2)))`
`Rightarrow f(x)={{:(,2sin^(-1)x,"if "-(1)/(sqrt2)le x le (1)/(sqrt2)),(,pi-2sin^(-1)x,"if "(1)/(sqrt2) le x le1),(,-pi-2sin^(-1)x,"if "-1le x le -(1)/(sqrt2)):}`
Clearly f(x) is continuous on [-1,1] but it is not differentiable at `x= pm (1)/(sqrt2)` such that
`f'(x)={{:(,(2)/(sqrt(1-x^(2))),"if "(-1)/(sqrt2)lt x lt (1)/(sqrt2)),(,(-2)/(sqrt(1-x^(2))),"if "(1)/(sqrt2)lt |x| lt 1):}`