Question

Let $f\left(x\right)=si{n}^{-1}\left(2x\sqrt{1-x^2}\right)$, then

• A. f(x) is continuous and differentiable at $x=\frac{1}{\sqrt{2}}$
• B. f(x) is continuous and differentiable at $x=\frac{\pi }{4}$
• C. f(x) is continuous and differentiable at $x=\frac{\pi }{6}$
• D. f(x) is continuous but non-differentiable at $x=-\frac{1}{\sqrt{2}}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B

f(x) is continuous and differentiable at $x=\frac{\pi }{4}$

C

f(x) is continuous and differentiable at $x=\frac{\pi }{6}$

D

f(x) is continuous but non-differentiable at $x=-\frac{1}{\sqrt{2}}$

$f\left(x\right)=si{n}^{-1}\left(2x\sqrt{1-x^2}\right)Putx=sin\theta \Rightarrow \theta =si{n}^{-1}xϵ[-\frac{\pi }{2},\frac{\pi }{2}]y=si{n}^{-1}\left(sin2\theta \right)$
$=\{\begin{array}{ccc}-(\pi +2si{n}^{-1}x)& ;& -1\le x<-\frac{1}{\sqrt{2}}\\ 2si{n}^{-1}x& ;& -\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}\\ (\pi -2si{n}^{-1}x)& ;& \frac{1}{\sqrt{2}}<x\le 1\end{array}$