Question

Find the number of points where the function $f\left(x\right)=(x^2-1)+\sin \left|x\right|$ is not differentiable.

Answer 1

Given function is $f\left(x\right)=x^2-1+\sin \left|x\right|$,

consider two cases,

case 1 : $x>0$,

$⟹f\left(x\right)=x^2-1+\sin x⟹f^{\prime }\left(x\right)=2x+\cos x⟹f^{\prime }\left(0\right)=1$

case 2 : $x<0$,

$⟹f\left(x\right)=x^2-1-\sin x⟹f^{\prime }\left(x\right)=2x-\cos x⟹f^{\prime }\left(0\right)=-1$

So here at point zero the left derivatives and the right derivaties are not equal.

So, the function is not differentiable at $0$, this means the graph may contain a kink, as the graph is continuous everywhere.