Consider the following procedure for solving the nonlinear equation f(x) = 0: Start with an initial guess x0.
For k = 0, 1, 2, . . . do the following:

a. Compute the value that standard Newton’s method would provide, and call it ˆxk+1, i. e.
xk+1 = xk- f(xk)/f'(xk)

b. Compute the next approximation xk+1 by averaging xk and xk+1, i. e.,

xk+1 = (xk + ˆxk+1) 2

c. Show that if this method converges, it will converge to a solution of f(x) = 0. (
d. Show that this method converges under the same conditions as Newton’s method.
e. Determine the order of convergence of this method