MATH 295 FINAL EXAM

IGOR KRIZ

Name: ID number:

Instructions: This exam is due on Tuesday December 9 at noon (at the beginning of class). Work individually. Collaboration on final exams is not allowed. Using books, notes or calculators for your own reference is OK, provided that full reasoning is given for all steps of your solution. (For example, if you graph a function, the graph can be used only if you prove all of its properties which you are using.) In your reasoning you may refer to theorems proven in class without proof. When writing up a solution, write down all the steps. The exam tests your ability to write up a solution, as well as your ideas. Plan on spending at least 2 hours of undisturbed time on this exam.

1. Prove that if I_{1} ⊇ I_{2} ⊇ I_{3} ⊇ ... are compact intervals, then their intersection is non-empty, i.e. there exists an x ∈ ℝ such that x ∈ I_{k} for all k = 1, 2, 3, ....

2. Prove that if f : ℝ → ℝ such that

lim_{x→0} f(x) = 0,

then

lim_{x→0} f(x) cos \frac{1}{x} = 0.

3. (a) Let a, b ∈ ℝ. Prove that the square root of the complex number a + bi is

\frac{1}{\sqrt{2}}\sqrt{\sqrt{a^{2} + b^{2}} + a} + \frac{1}{\sqrt{2}} \frac{ib}{\sqrt{\sqrt{a^{2} + b^{2}} + a}}.

(b) Using (a), find the roots of the polynomial

x^{2} + (1 + i)x + 1

in the form c + di, c, d ∈ ℝ.

4. Prove that the polynomial

f_{a}(x) = x^{3} − x^{2} + x + a

does not have two distinct real roots for any a ∈ ℝ. [Hint: use the mean value theorem.]

5. (a) Prove that if f : ℝ → ℝ satisfies f′(a) > 0, and f′(x) exists and is continuous at all x ∈ ℝ, then f is strictly increasing on (a − ϵ, a + ϵ) for some ϵ > 0. [Hint: use the mean value theorem.]

(b) Prove that the function f : ℝ → ℝ given by

f(x) = \{ \begin{matrix}\frac{1}{2}x + x^{2} sin \frac{1}{x} & for x ≠ 0 \\ 0 & for x = 0 \end{matrix}

satisfies f′(0) > 0, but is not increasing on any interval (−ϵ, ϵ) for ϵ > 0. [Hint: Consider the derivative.]