Math 3B
Summer Session A 2016
Challenge Problem 2
Due: 5 July
2016 Name: Using Calculus to understand a function
What would you do if I told you to convince me (without using wolfram alpha!) that the polynomial
5863 − 358094x + 111192x2 + 680600x3
has a root between 0 and 1? Probably skip this week’s challenge problem (or at least that’s what I’d
do, as that polynomial looks terrible). Instead this week we will look at a technique often used in
mathematics, where instead of answering a specific hard question, you generalize it until a pattern
is more visible, and it gets a bit easier. Part 1
Let a, b, c, d be real numbers which satisfy the following equation:
a+ b
c d
+ + =0
2 3 4 and consider the polynomial with them as coefficients, namely
f (x) = a + bx + cx2 + dx3
Convince me that f has a root in the interval (0, 1).
Hint: think about the relationship of functions with their derivatives and antiderivatives. What does
the mean value theorem say again?
How does this solve the question above? Part 2
Can this be generalized? Formulate a claim like the one above, except now for a polynomial of
degree N . What, if anything from the reasoning above must change to show that this polynomial
also has a root in the standard unit interval?
Note: In this problem you are trying to write up a convincing argument (really, a mathematical
proof ) instead of just doing a calculation. As such, your answer should be in full sentences / a
paragraph or two.