Problem 2:

(a) Use properties of quadratic functions to prove that 3x^2 (x + 1)^2 for all real x 4.

(b) Use mathematical induction and the inequality from part (a) to prove that 3n 2 n + 3n 2 for all integers n 4.

(c) Let g(n) = 2n + 3n 2 and h(n) = 3n. Using the inequality from part (b), prove that g(n) = O(h(n)). You need to give a rigorous proof derived directly from the definition of the O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how g(n) = O(h(n)) follows from this definition.)