To determine where the average rate of change of the function \( f(x) \) is the greatest, we need to analyze the slope of the secant line between two points on the graph of \( f(x) \). The average rate of change over an interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Without the image, I can't see the actual graph of \( f(x) \), but you can follow these steps to find the interval with the greatest average rate of change:
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Calculate the Average Rate of Change for each of the proposed intervals:
- For \([0,1]\): Compute \(\frac{f(1) - f(0)}{1 - 0}\).
- For \([1,4]\): Compute \(\frac{f(4) - f(1)}{4 - 1}\).
- For \([4,5]\): Compute \(\frac{f(5) - f(4)}{5 - 4}\).
- For \([5,6]\): Compute \(\frac{f(6) - f(5)}{6 - 5}\).
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Compare the values to find which interval has the highest average rate of change.
If you provide additional details about the graph or the function values at specific points, I could assist you further in calculating the average rates of change and finding the interval with the greatest value. Alternatively, if you see a noticeable steepness in the graph or a significant change in function value over an interval, that can also help you identify where the greatest average rate of change is likely occurring.
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