Question

The function $y={\sin }^{-1}\left(\cos x\right)$ is not differentiable at

• A. $x=\pi$
• B. $x=-2\pi$
• C. $x=2\pi$
• D. $x=\pi /2$

Answer 1

Answer: (A), (B), (C)

$x=-2\pi$
$x=2\pi$
$x=\pi$

Given, $y={\sin }^{-1}\left(\cos x\right)$

$\therefore y^{\prime }=-\frac{\sin x}{\sqrt{1-{\cos }^{2}x}}=-\frac{\sin x}{\sqrt{{\sin }^{2}x}}=-\frac{\sin x}{\left|\sin x\right|}$
So the function is not differentiable at the points where $\sin x=0$,
that is, for $x=k\pi (k\in I)$. In particular, $x=\pi ,2\pi ,-2\pi$.