($A$) differentiable at $x=0$ if $a=0$ and $b=1$
($B$) differentiable at $x=1$ if $a=1$ and $b=0$
($C$) $NOT$ differentiable at $x=0$ if $a=1$ and $b=0$
($D$) $NOT$ differentiable at $x=1$ if $a=1$ and $b=1$
A differentiable at $x=0$ if $a=0$ and $b=1$
$B$ differentiable at $x=1$ if $a=1$ and $b=0$
$f(x)=a \cos \left(\left|x^3-x\right|\right)+b|x| \sin \left(\left|x^3+x\right|\right)$
As we know that $\cos \theta=\cos (-\theta)$,
$f(x)=\left\{\begin{array}{ll}a \cos \left(x^3-x\right)-b x \sin \left(-x^3-x\right) & x < 0 \\ a \cos \left(x^3-x\right)+b x \sin \left(x^3+x\right) & x \geq 0\end{array}\right.$
$\therefore f ( x )= a \cos \left( x ^3- x \right)+ bx \sin \left( x ^3+ x \right) \quad \forall x \in R$
Hence, $f(x)$ is differentiable at all $x \in R$ for any $a$ and $b$.
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