Determine the smallest value for which the Newton method diverges if $\displaystyle f(x)=arctan(x)$.

My approach : Almost nothing. I know there's a theorem that states that if $\displaystyle f''(x)$ is continuous, $\displaystyle f$ is increasing, convex and has a zero, then this zero is unique and the Newton method will converges to it regardless of the starting point.
So I'm guessing that if I can show that $\displaystyle f''$ is not continuous on some points, or that arctan is not convex from some point (well, I'm guessing it, since I don't know if it's convex or not), maybe it would mean that the NM would diverges from this point. But I'm not sure it would mean so, since it's not a theorem nor a corollary. Anyone can help me? Maybe I'm missing an easy way to solve the problem.