What are the conditions for cubic convergence in functional iteration?

[DeadxBunny]

Homework Statement

Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

Homework Equations

I really don't know.

The Attempt at a Solution

I have no idea how to even begin!

Thanks in advance!

 

[Hurkyl]

I have no idea how to even begin!

Try starting with definitions.