In the derivation of Newton's method to determine the formula for x;+1 the function f(x) is approximated using a first order Taylor approximation centered at x;. This problem investigates what happens where you try to use a second order Taylor approximation. a) Approximating f(x) using a second order Taylor approximations centered at x; what is the resulting formula for x;+1? b) In theory a second order Taylor approximation should be more accurate than a first order Taylor approximation. However the formula in part (a) has several unpleasant complications that Newton's method doesn't have. Identify two of them. c) Given that x;+1 is close to x, what choice should be made for the + in part (a)? d) One way to avoid the complications considered in part (b) is to note that the Taylor approximation used in part (a) contains a term of the form (x,+1 - x;)?. Explain why this can be approximated with *7 . If this is done what is the resulting formula for x;+1? Note '(x) that the formula you are deriving is known as Halley's method, and it is an example of a third order method.