SCIENCE ONE: MATHEMATICS ASSIGNMENT 4 There are two parts to this assignment. The first part is online; you will need to register and login at www.mathxl.com. The second part consists of the questions on this page. You are expected to provide full solutions with complete arguments and justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled, with your name and student number at the top of each page. 1. Note that 1/2 1/4 1 1 = . 2 4 Prove that there are infinitely many pairs of distinct positive numbers a and b such that aa = bb . 2. Your textbook defines f to be concave up on an interval I if f 00 (x) > 0 for all x ∈ I, and concave down on I if f 00 (x) < 0 for all x ∈ I. If f changes concavity at c, c is called an inflection point. Prove the following two claims. (a) If f is concave up on I, then its graph lies above its tangent lines on that interval. (b) If f has an inflection point at c, then its graph crosses over its tangent line at c. 3. Write a question on the topic of “curve sketching” that would be reasonable to include on the December exam. Include the complete solution. The best question will be strongly considered for inclusion. If you have questions on what makes a good exam question, consult with your professors.