To solve this problem, we need to find the average rate of change of the function $f(x)$ over the interval $5 \leq x \leq 7$.

Step-by-Step Solution

1. Identify the Points:

   • Locate the values of $f(5)$ and $f(7)$ on the graph.
   • From the graph, approximate:
     • $f(5) \approx 30$
     • $f(7) \approx 20$

2. Use the Average Rate of Change Formula: The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$ Here, $a = 5$ and $b = 7$.

3. Plug in the Values: $\text{Average Rate of Change} = \frac{f(7) - f(5)}{7 - 5} = \frac{20 - 30}{2} = \frac{-10}{2} = -5$

Answer

The average rate of change of $f(x)$ on the interval $5 \leq x \leq 7$ is $-5$.

Would you like further details or have any questions?

Here are five related questions to explore further:

1. How would you interpret the significance of a negative average rate of change?
2. What would be the average rate of change over a different interval, such as $0 \leq x \leq 4$?
3. How does the concept of instantaneous rate of change differ from average rate of change?
4. Can the average rate of change be zero, and if so, under what conditions?
5. How would this problem change if we were asked for the average rate of change over a larger interval?

Tip: The average rate of change is simply the slope of the line connecting two points on a curve, making it a powerful tool for understanding the behavior of functions over intervals.