Let K be the set of all real values of x where the function f(x)=|x|-|x|+2(x-)|x| is not d…

Question

Let $K$ be the set of all real values of $x$ where the function $f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|$ is not differentiable. Then the set $K$ is equal to

Accepted Answer

For $x>0$: $f(x)=\sin x-x+2(x-\pi)\cos x$, $f'(x)=3\cos x-2(x-\pi)\sin x-1$ --- smooth for $x>0$. For $x<0$: $f(x)=-\sin x+x+2(x-\pi)\cos x$, $f'(x)=\cos x+1-2(x-\pi)\sin x$ --- smooth for $x<0$. At $x=0$: RHD $=3-1=2$; LHD $=1+1=2$ --- equal. So $f$ is differentiable everywhere, $K=\varnothing$.

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