Consider the function

f(x) = x³ + 2x² + 10x 20. In this lab, we will approximate a root r of the function f in a given interval using three dierent methods: the Bisectionmethod, Newtons method, and the Secant method.

Complete the following steps:

1. Show, using analysis, that the function f has a unique root in the interval [1, 2]; i.e., use techniques learned incalculus to show that

(i) there exists a root r of f in [1, 2], and

(ii) that there is only one root r of f in [1, 2].

2. Implement a MatLab function for each of the three methods. These routines should be very exible; i.e., you shouldbe able to apply them to functions other than the one posed in this project. You should write a separate MatLab function to evaluate the function f given above (you will also need a function for f to be used with Newtons method).

The program for each of the methods should accept a function f as an input (Newtons needs f as an inputas well).

The program for each of the methods should also have inputs for an error tolerance, a maximum numberof iterations, and any other information that the method needs to perform (initial guess x0 for Newtons,endpoints of the interval [a, b] for Bisection, and two initial guesses x0, x1 for the Secant method).

3. For this project, we will stop the iterations once our approximation to the root r is within an error of = 105. For the Bisection method, when the length of the current interval satises b a 2, taking the midpoint c

of [a, b] will satisfy |r c| . For the other two methods, stop the iterations once |xn+1 xn| .

4. For each method, format your output as follows: Output the name of the method. Output the starting information for each method (i.e., a and b for Bisection, x0 for Newtons, x0 and x1 for Secant).

Output a table for each method, organized into four columns, Include a header for the table.

Column 1: display the iteration index n,

Column 2: display the current approximation xn to the root r using 8 digits to the right of the decimal

, Column 3: display the value of f(xn), again using 8 digits to the right of the decimal,

Column 4: display the absolute error |r xn|, using 8 digits to the right of the decimal.

After each table, output the nal approximation to the root r, as well as the absolute error for the nalapproximation.