(a) Without doing a sketch, show that the cubic equation x 3 + x 2 + x 1 = 0 has at least one solution on the interval [0, 1]. [Hint: Use a theorem discussed in lectures, or see Section 1.8 of Calculus (7th ed) by Stewart.]

(b) Now, by sketching the cubic x 3 + x 2 + x 1, you should see that there is, in fact, exactly one zero in the interval [0, 1]. Use Newton's method to find this zero accurate to 3 decimal places. You should include a sketch of the cubic, Newton's iteration formula, and the list of iterates