(Solved): Please solve all parts. Consider the fixed point iteration method xn+1=1+4xn1. a. Show that ...

Consider the fixed point iteration method $x_{n+1}=1+\frac{1}{4x_n}$. a. Show that the method has a unique fixed point on $[1,2]$. b. Let $x_0=1$ and find $x_1$ and $x_2$. c. Then write down a polynomial equation that the fixed point(s) must satisfy and solve exactly for the fixed point(s). d. Find the order of convergence of the method to the root on $[1,2]$ and use your understanding of fixed-point method convergence to estimate by what factor the error should shrink at each step. e. Finally, use Aitken's method to improve on the values $x_0,x_1$ and $x_2$.