ASSIGNMENT 2·4 There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course webpage. 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 on the correctness, clarity and elegance of your solutions. Your answers must be typeset. They must be stapled, with your name and student number at the top of each page. 1. Let f (x) = 2x2 ln x − 13x2 . Find the critical point(s) of the function on its domain, and then use the Second Derivative Test to determine whether they correspond to local minima, local maxima, or neither. 2. Graph, and write down an algebraic expression for, a single function f with all of the following characteristics: (a) (b) (c) (d) the domain of f is (−∞, ∞), f 0 (x) > 0 for all x, f 00 (x) < 0 for all x, and there is no number M such that f (x) < M for all x. 3. The function 1 −t/2 te 4 is an example of a Gamma function, a kind of function used to model probable waiting times — for example, the probability of dying at a certain age, or of an action potential occurring at a certain time. In this case, suppose that the function P (t) models the probability P of a vireo, a small migratory bird, having a lifespan of t ∈ (0, ∞) years. P (t) = (a) What lifespan has associated with it the highest probability? What is that probability? (b) Sketch the graph of P (t), indicating its intervals of increase and decrease, as well as its concavity. (c) Make, and then defend in a few sentences, a conjecture about the total area under the graph which you sketched in part (b).