ASSIGNMENT 2·5
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) = x 12 − x. Determine its domain, roots, intervals of increase/decrease, local extrema, intervals
of concavity and inflection points, justifying your answers.
2. (a) Graph, and write down an algebraic expression for, four different functions, each of which satisfies
one of the following sets of criteria:
i. the function is increasing and concave up everywhere,
ii. the function is increasing and concave down everywhere,
iii. the function is decreasing and concave up everywhere, and
iv. the function is decreasing and concave down everywhere.
(b) Prove that a function f satisfying the condition
f 00 (x) > 0 everywhere
cannot also satisfy the condition
f (x) < 0 everywhere.
3. Plasma drug concentration curves, depicting the concentration of a drug in the bloodstream of a patient
with respect to time, play a major role in pharmacology. One such curve, taken from the 1996 paper
“Increased Plasma Rifabutin Levels with Concomitant Fluconazole Therapy in HIV-Infected Patients”
in Annals of Internal Medicine, is shown below. Let f (t) be the concentration of Rifabutin at time t,
represented by the dotted line.
(a) Approximately where does f attain a global maximum? What does this represent pharmacologically?
(b) It appears that the graph has an inflection point somewhere in the interval t ∈ (6, 18). What does
this inflection point represent?
(c) f (t) is given as a continuous function. Explain, in a few sentences, why this is a reasonable model.