Which statement correctly compares the two functions on the interval [-1, 2]?

Question

A.
Both functions are increasing, but function g increases at a faster average rate.
B.
Both functions are increasing, but function f increases at a faster average rate.
C.
Only function f is increasing, but both functions are negative.
D.
Only function f is increasing, and only function f is negative.

Answer 1

To determine the correct comparison between the two functions on the interval \([-1, 2]\), we need to analyze their behavior (whether they are increasing or decreasing) and their values over that interval. However, since the specific functions \(f\) and \(g\) are not provided, I cannot make a direct comparison.

However, I can guide you through the process for making this determination:

1. Identify the functions: Get the explicit forms of functions \(f(x)\) and \(g(x)\) defined on the interval \([-1, 2]\).

2. Calculate derivatives: Find the derivatives \(f'(x)\) and \(g'(x)\) to determine where each function is increasing or decreasing. A function is considered increasing on an interval if its derivative is positive throughout that interval.

3. Evaluate the signs of the functions: Check the values of each function between the limits of the interval \([-1, 2]\) to determine if they are negative or positive over that interval.

4. Average rate of increase: For functions that are increasing, you could find their average rates of increase by evaluating the differences \(f(2) - f(-1)\) and \(g(2) - g(-1)\), then dividing by the interval length (which is 3 for \([-1, 2]\)).

Using these steps, if you go through the actual calculation, this should help you identify which of the answer choices A, B, C, or D correctly describes the relationship between the two functions on the specified interval. If you have specific functions or values in mind, feel free to share, and I can provide more targeted assistance.