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