Using Newton's Method

10. (Gu Spring 2012 Sample Final #4) Consider the Newton's method 0Ck+1 = Ck f(Ik ) f' ( ICK ) k = 1, 2, ... for finding the root of the equation f(x) = 0. Assume that . * is a multiple root, i.e., f(x) = (x - x*)mg(x), where m > 1 is an integer and g(x) is a smooth function with g(x*) # 0. . Newton's method converges to a*. all its guard (d) Show that Newton's method must converge to x* linearly