Let $a, b \in \mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = a \cos(|x^{3} - x|) + b\,|x| \sin(|x^{3} + x|)$. Then $f$ is .

$f(x) = a\cos(|x^{3}-x|) + b\,|x|\sin(|x^{3}+x|)$ $a, b \in \mathbb{R}$ $\text{Check differentiability at } x=0 \text{ and at } x=1$ $\text{for the } (a,b) \text{ given in each option}$ $\cos(-t) = \cos t \Rightarrow \cos(|x^{3}-x|) = \cos(x^{3}-x)$ $\Rightarrow a\cos(|x^{3}-x|) \text{ is smooth, hence differentiable everywhere}$ $|x^{3}+x| = |x|\,(x^{2}+1)$ $b\,|x|\sin(|x^{3}+x|) = b\,|x|\sin\!\big(|x|(x^{2}+1)\big)$ $b\,|x|\sin\!\big(|x|(x^{2}+1)\big) = b\,x^{2}(x^{2}+1) - \dfrac{b\,x^{4}(x^{2}+1)^{3}}{6} + \cdots$ $f(x) = a + \left(b - \dfrac{a}{2}\right)x^{2} + O(x^{4})$ $\Rightarrow f'(0) = 0 \text{ exists for every } a, b$ $x=1:\quad |x|=x,\quad |x^{3}+x| = x^{3}+x \ (\text{smooth})$ $f'(1) = b\,(4\cos 2 + \sin 2) \text{ exists for every } a, b$ $f \text{ is differentiable at } x=0 \text{ and at } x=1 \text{ for ALL } a, b$ $(1)\ \checkmark \qquad (2)\ \checkmark \qquad (3)\ \times \qquad (4)\ \times$ $\boxed{\text{Answer: options (1) and (2)}}$