Consider f(x) = 3x3 - 2x + 3x - 2 over the interval 0 x 4....

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Consider f(x) = 3x3 - 2x + 3x - 2 over the interval 0 x 4. a. What is the exact root of in this interval? Hint: Factor by grouping. b. In what interval is the root guaranteed to be after three iterations of the bisection method? c. Based on your answer to (b), what is the best guess for the root of f after three iterations of the bisection method? d. What is the maximum possible error of your answer to (c)? e. How many iterations of the bisection method are needed to guarantee that the maximum possible error of the best guess of the root of f is less than 10-6? Consider again f(x) = 3x 2x + 3x 2. a. Perform one iteration of Newton's method with initial guess x0 = 1. b. A grad student is concerned that there is an initial guess x such that f'(x) = 0 for some k 0. In such a situation, Newton's method will fail. Explain why the grad student has nothing to worry about. Hint: Consider the values of x such that f'(x) = 0. c. Perform one iteration of the secant method with initial guesses x0 = 0 and x = 1. Consider f(x) = 3x3 - 2x + 3x - 2 over the interval 0 x 4. a. What is the exact root of in this interval? Hint: Factor by grouping. b. In what interval is the root guaranteed to be after three iterations of the bisection method? c. Based on your answer to (b), what is the best guess for the root of f after three iterations of the bisection method? d. What is the maximum possible error of your answer to (c)? e. How many iterations of the bisection method are needed to guarantee that the maximum possible error of the best guess of the root of f is less than 10-6? Consider again f(x) = 3x 2x + 3x 2. a. Perform one iteration of Newton's method with initial guess x0 = 1. b. A grad student is concerned that there is an initial guess x such that f'(x) = 0 for some k 0. In such a situation, Newton's method will fail. Explain why the grad student has nothing to worry about. Hint: Consider the values of x such that f'(x) = 0. c. Perform one iteration of the secant method with initial guesses x0 = 0 and x = 1.