Final answer:
Yes, it is possible for a function to have portions that increase in a given interval and still have a negative average rate of change for the entire interval. The opposite scenario is not possible.
Step-by-step explanation:
Yes, it is possible for a function to have portions that increase in a given interval and still have a negative average rate of change for the entire interval. Let's consider an example:
Suppose we have a function f(x) that has two parts. Part A begins with a nonzero y-intercept and has a downward slope that levels off at zero. Part B begins at zero with an upward slope that decreases in magnitude until the curve levels off.
In this scenario, the function has portions that increase in a given interval (Part B), but because Part A has a negative slope, the overall average rate of change for the entire interval could still be negative.
On the other hand, it is not possible for a function to have portions that increase in a given interval and have a negative average rate of change for the entire interval. For the opposite scenario, let's consider an example where Part A begins with a nonzero y-intercept and has an upward slope that levels off at zero, and Part B begins at zero with an upward slope that increases in magnitude until it becomes a positive constant. In this case, both parts of the function have positive slopes, so the overall average rate of change for the entire interval would also be positive.