Question

Number of points where $f\left(x\right)=\cos \left|x\right|+\left|\sin x\right|$ is not differentiable in $x\in (0,4\pi )$, is:

Answer 1

For $x\in (0,4\pi )$,
$\left|x\right|=x\Rightarrow f\left(x\right)=\cos x+\left|\sin x\right|$
Now,
$\cos x$ is continuous and differentiable everywhere for $x\in (0,4\pi )$, but $|\sin x|$ is non differentiable for $x=\pi ,2\pi ,3\pi$ because at these points, it has sharp corners.
So, $f\left(x\right)$ is not differentiable at $3$ points.