Question

Number of points where $f\left(x\right)=\left|\cos \left|x\right|\right|+{\cos }^{-1}\left(sgnx\right)+\left|{\log }_{e}x\right|$ is not differentiable in $(0,2\pi )$ is

Answer 1

$f\left(x\right)=\{\begin{array}{c}\cos x+{\cos }^{-1}1-{\log }_{e}x;0<x\le 1\\ \cos x+{\cos }^{-1}\left(1\right)+{\log }_{e}x;1<x\le \pi /2\\ -\cos x+{\cos }^{-1}\left(1\right)+{\log }_{e}x;\pi /2<x\le 3\pi /2\\ \cos x+{\cos }^{-1}\left(1\right)+{\log }_{e}x;3\pi /2<x\le 2\pi \end{array}$

at $x=1$

${lim}_{x\to 1^{-}}f\left(x\right)=\cos 1+{\cos }^{-1}1-0$

${lim}_{x\to 1^{+}}f\left(x\right)=\cos 1+{\cos }^{-1}1+0$

LHL$=$RHL $\Rightarrow$continuous

at $x=\pi /2$

${lim}_{x\to {\pi /2}^{-}}f\left(x\right)=\cos \pi /2+{\cos }^{-1}\left(1\right)+{\log }_e\pi /2$

$={\cos }^{-1}\left(1\right)+{\log }_e\pi /2$

${lim}_{x\to {\pi /2}^{+}}f\left(x\right)=-\cos \pi /2+{\cos }^{-1}\left(1\right)+{\log }_e\pi /2$

$={\cos }^{-1}\left(1\right)+{\log }_e\pi /2$

LHL$=$RHL $\Rightarrow$continuous

Similarly it is continuous at $x=3\pi /2$

$\Rightarrow f\left(x\right)$ is continuous over the interval $(0,2\pi )$

$f^{{}^{\prime }}\left(x\right)=\{\begin{array}{c}-\sin x-1/x;0<x\le 1\\ -\sin x+1/x;1<x\le \pi /2\\ \sin x+1/x;\pi /2<x\le 3\pi /2\\ -\sin x+1/x;3\pi /2<x\le 2\pi \end{array}$

at $x=0$

${lim}_{x\to 1^{-}}{f}^{\prime }\left(x\right)=-\sin 1-1$

${lim}_{x\to 1^{+}}{f}^{\prime }\left(x\right)=-\sin 1+1$

LHL$\ne$RHL $\Rightarrow$Non differentiable

at $x=\pi /2$

${lim}_{x\to {\pi /2}^{-}}{f}^{\prime }\left(x\right)=-1+2/\pi$

${lim}_{x\to {\pi /2}^{+}}{f}^{\prime }\left(x\right)=1+2/\pi$

LHL$\ne$RHL $\Rightarrow$Non differentiable

at$x=\pi /2$

${lim}_{x\to {3\pi /2}^{-}}{f}^{\prime }\left(x\right)=-1+2/3\pi$

${lim}_{x\to {3\pi /2}^{+}}{f}^{\prime }\left(x\right)=1+2/3\pi$

$\Rightarrow$Non differentiable

$\therefore f\left(x\right)$ is non differentiable at $3$ points.