To determine the interval over which the average rate of change is greatest, we would typically analyze the graph of the function \( f(x) \) and look for the steepest slope between the given intervals. However, since I cannot see the image provided, I can't calculate or evaluate the rates of change directly.
That said, the average rate of change for a function over an interval \([a, b]\) is calculated using the formula:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
To find the interval with the greatest average rate of change, you would evaluate this expression for each of the given intervals ([0,1], [1,4], [4,5], and [5,6]) using the values of \( f \) at the endpoints of these intervals.
You can look at the intervals and calculate them, and the one with the highest value will be the interval over which the average rate of change is greatest.
Please provide an analysis of the slopes for each interval based on the graph or provide the values of \( f(x) \) at the specified points so I can help you determine the correct answer.