To determine on which interval the average rate of change of the function \( f(x) \) is the greatest, we need to analyze the changes in the values of \( f(x) \) over the specified intervals.
The average rate of change of a function over an interval \([a, b]\) is calculated using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
If you have the graph of the function, look for areas where the function rises or falls the most sharply, as these will correlate to greater average rates of change.
Based on the provided intervals:
- [4, 5]
- [0, 1]
- [1, 4]
- [5, 6]
You'd need to evaluate these intervals using the formula provided.
- Check \( f(4) \) and \( f(5) \) for the interval [4, 5].
- Check \( f(0) \) and \( f(1) \) for the interval [0, 1].
- Check \( f(1) \) and \( f(4) \) for the interval [1, 4].
- Check \( f(5) \) and \( f(6) \) for the interval [5, 6].
Without seeing the graph, I can't tell you exactly which interval has the greatest rate, but follow these steps to calculate it using the points you have on the graph. The option with the highest result from your calculations will indicate the interval with the greatest average rate of change.
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