7. (bonus) Let the function f : R - R be strictly increasing and strictly convex, e.g both the first and second derivatives are positive on R. Assume that f(0) = 0, f(-1) 0. Define the following two sequences (Xn, Un) : x(0) = 1, y(0) = -1; En+1 = En - f(x ) (1) ; Un+1 = Un- f(an) f (yn) f ( In ) ( 1 ) . Prove that 0 Un+1 > Un. To what limit and how fast do they converge?8. Suppose that an\" S be?\" GO > 0. Prove that an 3 %(ba0)2n. What this bound has to do with the Newton Method