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On which interval is the average rate of change the greatest?

[–3, –2]

 [–2, –1]

 [2, 3]

 [5, 6]

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.

On which interval is the average rate of change the greatest?

Explore how to determine which interval has the greatest average rate of change for a given function based on its graph. Using the formula (f(b) - f(a)) / (b - a), this problem guides you through analyzing various intervals on a function's graph. Suitable for students in Grades 11-12.

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