To determine the interval where the average rate of change is the greatest based on the described graph of \( f(x) \), we calculate the average rate of change for each interval mentioned in the options:
-
Interval [0, 1]:
- Calculate \( f(0) \) and \( f(1) \).
- Average rate of change = \( \frac{f(1) - f(0)}{1 - 0} \).
-
Interval [1, 4]:
- Calculate \( f(1) \) and \( f(4) \).
- Average rate of change = \( \frac{f(4) - f(1)}{4 - 1} \).
-
Interval [4, 5]:
- Calculate \( f(4) \) and \( f(5) \).
- Average rate of change = \( \frac{f(5) - f(4)}{5 - 4} \).
-
Interval [5, 6]:
- Calculate \( f(5) \) and \( f(6) \).
- Average rate of change = \( \frac{f(6) - f(5)}{6 - 5} \).
To identify where the average rate of change is greatest, the two key intervals likely experiencing rapid vertical changes are [4, 5] and [5, 6]. Intervals with steeper slopes will yield greater average rates of change.
Based on your description that the function goes up notably to 1, dips back down notably to 4 and then rises again to 6, we can infer:
- The interval [4, 5] likely experiences a large increase in \( f(x) \) relative to a small increase in \( x\).
- The interval [5, 6] would continue that increase.
Given the options, it is most likely that the greatest average rate of change is for the interval [5, 6], where the function appears to rise sharply.
So, the correct response would be:
The function has the greatest average rate of change over the interval [5, 6].
1 answer