Question

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

1) f′(x) = 0
2) f'(x) = 0
3) Cannot be determined

Answer 1

For a polynomial function with degree greater than 2, at least one value of x between a and b must have a positive slope that is decreasing in magnitude. Therefore, option 1) may or may not be true, but option 2) must be true.

Step-by-step explanation:

The question asks which of the following must be true for at least one value of x between a and b, given that f(a) is positive and f'(a) is negative. We are given two options: 1) f′(x) = 0 and 2) f'(x) > 0. Let's analyze each option:

1. Initially, f'(a) is negative, so for there to be a value of x between a and b such that f′(x) = 0, f'(x) must change sign from negative to positive at some point. This means the function must have a local minimum between a and b. However, since the function has a degree greater than 2, it may also have local maxima or additional inflection points between a and b. Therefore, option 1) is not necessarily true.
2. If f'(x) > 0 for all x between a and b, this means the slope of the function is always positive. Since f(a) is positive, this implies the function is strictly increasing between a and b. Therefore, option 2) is necessarily true for at least one value of x between a and b.

In conclusion, option 1) may or may not be true, but option 2) must be true for at least one value of x between a and b.