Can someone help me with this problem?
Consider the fixed point iteration
a) Show that the iteration converges for any initial guess satisfying
b) Assume that the iteration converges, find m such that the method converges quadratically.
I think I got part a) by letting , then take the derivative . I then show that there exists such that , and by a theorem this implies the iteration converges for any initial guess in the specified interval.
For part b), I need to find m such that exists, but I don't get anywhere from here. Hope someone can give a hand.