Determine the root of the function f(x) = x 2e^(-x), with five significant figures, using the procedures listed below.

Part One (30 Points): Carry out the following requirements by hand. In each question, show the calculations for each iteration.

(a) Using the bisection method. Start with x1 = 0 and x2 = 1, and carry out the first three iterations.

(b) Using the secant method. Start with x1 = 0 and x2 = 1, and carry out the first three iterations.

(c) Using Newton-Raphson method. Start at x1 = 1 and carry out the first three iterations.

(d) Using the false-position method. Start with the two points, x1 = 0 and x2 = 1, and carry out the first three iterations.

(e) Using fixed-point iteration method, Start at x1 = 1 and carry out the first three iterations. Before that, check that the method will eventually converge with the initial guess.

Part Two (20 Points): Carry out the following requirements by MATLAB. In each question, along with the output from MATLAB, provide the complete code that you used.

(f) Find and use a MATLAB built-in function; find the root of the function. Also, please provide a brief description of the methods used by this function (Hint: use the MATLAB help tool).

(g) Write a MATLAB user-defined function that solves the nonlinear function f(x) = 0 using the bisection method. Use this function to find the root of the equation with initial points x1 = 0 and x2 = 1 and 10 iterations. The output should be a matrix, having the results of each iteration.

(h) Write a MATLAB user-defined function that solves the nonlinear function f(x) = 0 using Newton-Raphson method. Use this function to find the root of the equation with initial points x1 = 1 and 10 iterations. The output should be a matrix comprising the results of each iteration.

(i) Make a plot of Tolerance in f(x) Vs. Iteration Number. The figure should be one plot that include the results from (g) and (h) and has the necessary labels and legend.