To solve this problem, we need to find the average rate of change of the function $f(x)$ over the intervals provided in the options. The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula:

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

Let's go through each interval:

1. Interval [0, 4] (A):

   • $f(0) = -2$ (from the graph)
   • $f(4) = 6$ $\frac{f(4) - f(0)}{4 - 0} = \frac{6 - (-2)}{4} = \frac{8}{4} = 2$ Average rate of change: 2.

2. Interval [0, 2] (B):

   • $f(0) = -2$
   • $f(2) = 8$ $\frac{f(2) - f(0)}{2 - 0} = \frac{8 - (-2)}{2} = \frac{10}{2} = 5$ Average rate of change: 5.

3. Interval [2, 4] (C):

   • $f(2) = 8$
   • $f(4) = 6$ $\frac{f(4) - f(2)}{4 - 2} = \frac{6 - 8}{2} = \frac{-2}{2} = -1$ Average rate of change: -1.

4. Interval [2, 5] (D):

   • $f(2) = 8$
   • $f(5) = 0$ $\frac{f(5) - f(2)}{5 - 2} = \frac{0 - 8}{3} = \frac{-8}{3} \approx -2.67$ Average rate of change: approximately -2.67.

Now, let's order them from least to greatest:

• D: $-2.67$
• C: $-1$
• A: $2$
• B: $5$

Thus, the correct order is: D, C, A, B

Would you like any further details or have any questions?

Relative questions:

1. How is the average rate of change related to the slope of a secant line?
2. What does a negative average rate of change indicate about the function's behavior?
3. How does the average rate of change differ over smaller and larger intervals?
4. Can you explain why the interval [0, 2] has a larger rate of change than [0, 4]?
5. How would the interpretation of the rate of change change if the function were not continuous?

Tip:

The average rate of change over an interval can give insight into the overall trend of the function within that range, such as whether it is increasing or decreasing.