To determine the interval where the average rate of change of the function $f(x)$ is the greatest, we need to assess how the function behaves on each of the given intervals. The average rate of change over an interval $[a, b]$ is computed using the formula:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

We will analyze each of the given intervals from the graph:

1. Interval [–3, –2]:

   • $f(-3) \approx 1.5$
   • $f(-2) \approx -0.5$
   • The average rate of change is:
     $\frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{-0.5 - 1.5}{1} = -2$

2. Interval [–2, –1]:

   • $f(-2) \approx -0.5$
   • $f(-1) \approx -1.5$
   • The average rate of change is:
     $\frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{-1.5 - (-0.5)}{1} = -1$

3. Interval [2, 3]:

   • $f(2) \approx 4$
   • $f(3) \approx 6$
   • The average rate of change is:
     $\frac{f(3) - f(2)}{3 - 2} = \frac{6 - 4}{1} = 2$

4. Interval [5, 6]:

   • $f(5) \approx 9$
   • $f(6) \approx 4$
   • The average rate of change is:
     $\frac{f(6) - f(5)}{6 - 5} = \frac{4 - 9}{1} = -5$

Conclusion:

The interval with the greatest average rate of change is $[2, 3]$, where the rate of change is $2$.

Let me know if you'd like further clarification or have any questions. Here are five related questions you can consider:

1. How is the average rate of change of a function related to its derivative?
2. What is the physical meaning of the average rate of change in real-world applications?
3. How would you find the instantaneous rate of change at a specific point?
4. Could the average rate of change be positive on some intervals and negative on others? Why?
5. What is the relationship between increasing/decreasing intervals and the sign of the average rate of change?

Tip: The average rate of change between two points is essentially the slope of the secant line that connects those two points on the graph.