Fixed Point Iteration - Divergence to Convergence
Please refer to the question in the image.
My attempts are as follows:
1) Sub in g(x)=x for a final result of h(x)=x, then x=alpha obtains the result, but not sure if there is more to it.
2) I am not sure as by (1) my result is simply h(alpha)=alpha, which means c=all real numbers (so I must be going in the wrong direction already)
3) I know for quadratic convergence h'(alpha)=0 and h''(alpha) does not equal zero, but I cannot get this due to (2)
Thank you for your help
hint
$$h'(x)=1+(g'(x)-1)c $$
It converges if
$$-1 <h'(\alpha)<1$$
or
$$-1 <1+(g'(\alpha)-1)c <1$$ which gives
$$\frac{-2}{g'(\alpha)-1}<c <0$$
For the quadratic convergence, we need $$h'(\alpha)=0$$ which means $$c=\frac {1}{1-g'(\alpha)} $$