Consider the function:
$$f(x) = \begin{cases} -1 + \arctan x & x < 0 \\ ax + b & x \geq 0 \end{cases}$$
Determine for which values of the real parameters $a, b$ the function is differentiable. Establish whether there exists an interval of $\mathbb{R}$ in which the function $f$ satisfies the hypotheses of Rolle's theorem. Justify your answer.