Question

Let f be a polynomial function with degree greater than 2. If a ‡ b and f(a) = f(b) = 1, which of the following must be true for at least one value of x between a and b

I. f(x) = 0
II. f'(x) = 0
III. f"(x) = 0

a. I only
b. II only
c. I and II only
d. I, II, and III

Answer 1

B
Rolle's Theorem

I and III are false - consider (x^2-1)^2 + 2/3 on [√(1 - 1/√3),√(1 + 1/√3)]
f" = 0 at x = 1/√3, but that is outside [a,b]

Answer 2

Just in case you are wondering, I started with f(x) = 1-x^4, but that has f"(0) = 0. So, I considered a more bumpy (x-1)^2 * (x+1)^2 = (x^2-1)^2. That has f(0) = f(2) = 1, but it also has f"(1/√3) = 0, which is in the interval [0,2].
So, I shifted it up until 1/√3 was outside [a,b]