ASSIGNMENT 1·9
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 or very neatly written. They must be stapled, with your name
and student number at the top of each page.
1. (a) Consider the polynomial
f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0
where an , an−1 , . . . , a1 , a0 are coefficients and an 6= 0. What is the maximum number of points on
the graph of f (x) with horizontal tangent lines? Justify your answer.
(b) Let g(x) = 3x4 + 20x3 + 18x2 − 108x. Find all the points on the graph of g(x) with horizontal tangent
lines.
2. Can a polynomial ever have a vertical tangent line? Justify your answer.
3. Find a general formula for the derivative
d
((x − c1 )(x − c2 )(x − c3 ) . . . (x − cn ))
dx
where c1 , c2 , c3 , . . . , cn are constants.. Justify your answer.