is anyone can do Math for master degree?

MATH 451, Fall 2014: Test 2 (take-home part) NAME (Print):

The solutions to the problems below are due Thursday, November 20 at 11:30am in class (ARM 117).

This exam is to be worked on strictly individually! No cooperation of any kind is allowed! The instructor will not provide any hints; you may only contact the instructor if further clarification of a certain statement is necessary.

Please print out this PDF file and show your work (if possible) in the space allotted. Staple to this (in the right sequence) whatever extra sheets of paper are needed.

Good luck!

1. (10 points) Let L ∈ R and let {an}n be a sequence of real numbers such that all its monotone subsequences converge to L. Prove that {an}n converges to L. Is the converse of this statement true (formulate and prove it if so)?

2. (a) (3 points) Write a mathematical definition of what it means for a function f : R → R to “be negative for sufficiently large |x|”.

(b) (7 points) Let f : R → R be a continuous function such that f(0) = 1 and such that f(x) is negative for sufficiently large |x|. Prove that there exist at least two distinct real numbers a such that f(a) = a2.

3. (a) (8 points) Prove that if f, g : R → R are bounded and uniformly continuous, then the product function fg is also uniformly continuous on R.

(b) (2 points) Is the above statement still true if we drop the boundedness assumption?

4. (a) (2 points) Find the points (if any) where the function f(x) = |x| is not differentiable. Justify!

(b) (5 points) Use the definition of derivative to compute f ′(x) if f(x) = x|x| for all x ∈ R.

(c) (3 points) Use (b) and the product rule to show that if n ∈ N, then the function g(x) = xn|x| is differentiable everywhere on R. Compute g′.