The function $f(x)=0$ has a simple root in the interval $(1,2)$. The function $f(x)$ is such that $|f(x)|>3$ and $|f''(x)|\leq 4$ for all $x\in (1,2)$. Assuming that the Newton-Raphson method converges for all initial approximations in $(1,2)$, find the maximum number of iterations required to obtain the root correct upto 6 decimal places after rounding.
I know that Newton-Raphson method converges if $|f(x)f''(x)|<|f'(x)|^2$. I don't how to use it here and how to get the maximum number of iterations required. Please help.