22 1. Let f(x) = 2 2 - 9 (a) Determine the domain of the function (b) Determine the x and y intercepts. (c) Determine any asymptotes of the function. (d) Determine the intervals on which the f is increasing and decreasing, and identify any relative maximum or minimum values. Justify your answer. 2. Suppose f'(x) = 2x, and that the graph of f passes through the point (1, 0). Determine find a function that has the same derivative. Justify your answer. 3. In this problem we will use Rolle's Theorem to demonstrate that certain polynomials only have one root. (a) What is the conclusion of Rolle's Theorem given any two points such that f(a) = f(b) = 0. (b) Let p(x) = x2n+1 + ax + b for a > 0 i. Will p' (x) ever equal 0? Explain your answer. ii. Prove that p cannot have two real roots. To this end, assume that there are two real roots, T1 and T2. What must be true if they are indeed real roots? Hint: Use your answer to part (a) (c) Show that ' + x3 + x + 1= 0 has exactly one solution