The question states:

"A function $f(x)$ has a positive average rate of change on the interval $[1, 4]$, no average rate of change at all on the interval $[4, 6]$, and a negative average rate of change on the interval $[6, 9]$. Which of the following graphs best models this piecewise function?"

Analysis of Conditions

1. Positive Average Rate of Change on $[1, 4]$:

   • The function should be increasing from $x = 1$ to $x = 4$.

2. No Average Rate of Change on $[4, 6]$:

   • This suggests a constant function between $x = 4$ and $x = 6$, where the function does not change and has a slope of zero.

3. Negative Average Rate of Change on $[6, 9]$:

   • The function should be decreasing from $x = 6$ to $x = 9$.

Graph Analysis

Looking at the graphs:

• First Graph (Option A):

  • This graph shows an increasing section, a flat (constant) section, and a decreasing section, which matches the requirements of the problem statement.

• Other Options:

  • The remaining graphs either do not have a clear constant section between $x = 4$ and $x = 6$ or do not properly match the positive-to-constant-to-negative trend described.

Conclusion

The first graph (Option A) best models the function $f(x)$ as described in the problem.

Would you like further details or have any questions about this solution?

Here are some related questions you might consider:

1. How is the average rate of change calculated over an interval?
2. What other intervals could show different behaviors, like concavity changes, without affecting the rate of change sign?
3. How would a positive rate of change over the entire interval affect the graph's overall shape?
4. What if the constant rate of change interval was shifted—how would that affect the graph selection?
5. How can we distinguish between a zero average rate of change and no rate of change at all?

Tip: When analyzing intervals for a function’s behavior, sketching the graph based on the interval behavior can help visualize and choose the correct model more accurately.