Problem 2. Find an approximation to a root of x3 +x4 = 0.

2a) Show that x3 +x4 has at least one root in the interval [1,4].

2b) Show that derivative of x3 + x 4 is strictly positive for all x R. Use this observation to show x3 + x 4 has only one real root.

2c) Write Matlab programs for the two methods (Bisection method and Newton method) based on the model program given below. For Newtons methods, use the initial guess x0 = 1. Print out the approximation given after each iteration and the total number of iterations required by each method. Note that for these methods, we will assume we have satisfied the given accuracy requirement if two successive iterates agree to the given error tolerance, i.e., |xn+1 xn| 103.