When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is there any special strategies to get the $g(x)$ which will converge to the real root rapidly? I have to find the root between $[1,2]$ interval of the following function. $$ f(x) = x^3 + 4x^2 - 10 $$ I can get following different functions to do the fixed point iteration. $$g_1(x) = x - x^3 - 4x^2 +10$$ $$g_2(x) = (\frac{10}{x} - 4x)^\frac{1}2$$ $$ g_3(x) = \frac{1}2(10-x^3)^\frac{1}2$$ $$g_4(x) = (\frac{10}{4+x})^\frac{1}2$$ $$g_5(x) = x - \frac{x^3 + 4x^2 - 10}{3x^2 + 8x}$$
After checking each function for the first derivative, $$g_1(x),g_2(x),g_3(x) > 1$$ and $$g_5(x) < g_4(x)< 1.$$
Now I can see that the $g_5(x)$ is the suitable function for the iteration.
How can I get the proper $ g(x)$ without solving the derivative for all the possibilities?