Question

Let $f\left(x\right)=xco{s}^{-1}\left(-sin|x|\right),xϵ\left[-\frac{\pi }{2},\frac{\pi }{2}\right],$ then which of the following is true ?

• A. $f^{\prime }\left(0\right)=-\frac{\pi }{2}$
• B. f' is decreasing in $\left(-\frac{\pi }{2},0\right)$ and mcreasimng in $\left(0,\frac{\pi }{2}\right)$
• C. f is not differentiable at $x=0$
• D. f' is increasing in $\left(-\frac{\pi }{2},0\right)$ and decreasing in $\left(0,\frac{\pi }{2}\right)$

Answer 1

Answer: (B)

$f(x)$ is an odd function.
Npw, if $x\ge 0,f\left(x\right)=xco{s}^{-1}\left(-sinx\right)$
$=x\left(\frac{\pi }{2}-si{n}^{-1}\left(-sinx\right)\right)=x\left(\frac{\pi }{2}+x\right)$
Hence, $f(x)=\{\begin{array}{ll}x\left(\frac{\pi }{2}+x\right);& \text{x∈[0,2π]}\\ x\left(\frac{\pi }{2}-x\right);& \text{ x∈−[2π,0) }\end{array}$
so, $f(x)=\{\begin{array}{ll}\frac{\pi }{2}+2x;& \text{x∈[0,2π)}\\ \frac{\pi }{2}-2x;& \text{ x∈(−2π,0) }\end{array}$