Mathematics 242 – Principles of Analysis Problem Set 8, Due: 11/11

I. From Abbott: 4.4.2, 4.5.2, 4.5.3, 4.5.7, 5.2.3, 5.2.4.

Additional Problems

II. A) Show that if f(x) is any polynomial of odd degree, then f has a real root (that is a

real number c where f(c) = 0). B) Suppose that f(x) is a polynomial of even degree and that there exist a ≠ b in R such

that f(a) < 0 < f(b). Show that f has at least two distinct real roots. C) True or False: Any polynomial of even degree with a real root has at least two distinct

real roots.

III. Show that the equation x^{2} = 3^{x} has at least one real solution.

IV. A) Suppose f is continuous on [0, 1] with f(0) < 0 and f(1) > 1. Prove that there is at

least one point c ∈ (0, 1) where f(c) = c^{2}. B) Generalize your reasoning from part A to show that if f is as before and g is any

continuous function on [0, 1] with g(0) ≥ 0, g(1) ≤ 1, the there is at least one point c ∈ (0, 1) where f(c) = g(c).