numerical methods

please write very detailed solution, pen and paper question. please do all parts.

please don't attempt the question if you are not good in numerical methods... please please

4. Derive an iterative method by considering the Taylor series ap- proximation to the function, retaining terms up to the second derivative. The resulting quadratic equation can be solved to give the iteration formula Xi+1 = Xi 2f(xi) f'(xi) = V f'(xi)2 2f(xi)f" (xi) 1 where the sign of square root is chosen such that the denomi- nator has larger magnitude. Show that this iteration converges cubically to a simple root. What happens for multiple roots? 4. Derive an iterative method by considering the Taylor series ap- proximation to the function, retaining terms up to the second derivative. The resulting quadratic equation can be solved to give the iteration formula Xi+1 = Xi 2f(xi) f'(xi) = V f'(xi)2 2f(xi)f" (xi) 1 where the sign of square root is chosen such that the denomi- nator has larger magnitude. Show that this iteration converges cubically to a simple root. What happens for multiple roots