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The graph of the function f and a pointed label A no the graph are shown. Which of the following statements about the graph of F at point A is true? A. At point a the average rate of change of F is negative and the graph of F is concave down B. At point a the average rate  of change is negative and the graph of F is concave up. C. At point an average rate of change of F is positive and the graph of F is concave down. D. Point the rate of change of F positive and the graph of F is concave up.

To determine which statement about the graph of \( f \) at point \( A \) is true, we need to consider both the slope of the tangent at \( A \) and the concavity of the graph at that point.

### Observations at point \( A \):
1. **Rate of change (Slope)**: The slope of the tangent at \( A \) determines the rate of change. From the graph, the tangent at \( A \) appears to be slightly negative, meaning the function is decreasing at \( A \). So, the rate of change at \( A \) is negative.
   
2. **Concavity**: To determine concavity, we examine whether the curve is "bending" upwards or downwards at \( A \). The graph at \( A \) seems to be curving downward, which indicates concave **down** behavior.

### Now let's evaluate the options:
- **A.** "At point A, the average rate of change of \( f \) is negative and the graph of \( f \) is concave down."  
  This statement seems to match our observations: the rate of change is negative and the graph is concave down at \( A \).  

- **B.** "At point A, the average rate of change is negative and the graph of \( f \) is concave up."  
  The rate of change is correct (negative), but the graph is concave down, not up.

- **C.** "At point A, the average rate of change of \( f \) is positive and the graph of \( f \) is concave down."  
  The graph is concave down, but the rate of change is not positive; it's negative.

- **D.** "At point A, the rate of change of \( f \) is positive and the graph of \( f \) is concave up."  
  Both parts of this statement are incorrect, as the rate of change is negative and the graph is concave down.

### Conclusion:
The correct answer is **A**: "At point A, the average rate of change of \( f \) is negative and the graph of \( f \) is concave down."

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Would you like more details or have any further questions? Here are 5 related questions for you to consider:

1. How can you calculate the exact rate of change at a point using derivatives?
2. What are the different ways to determine concavity on a graph?
3. How does the second derivative test relate to determining concavity?
4. How can we find critical points on a function graph, and how do they relate to increasing/decreasing behavior?
5. What role does inflection points play in changing concavity?

**Tip:** To determine the concavity of a function analytically, you can use the second derivative. If the second derivative is positive, the function is concave up, and if it's negative, the function is concave down.

The graph of the function f and a pointed label A on the graph are shown. Which of the following statements about the graph of f at point A is true? A. At point A the average rate of change of f is negative and the graph of f is concave down B. At point A the average rate of change is negative and the graph of f is concave up. C. At point A the average rate of change of f is positive and the graph of f is concave down. D. At point A the rate of change of f is positive and the graph of f is concave up.

Explore how to determine the rate of change and concavity at a specific point on a function graph. Using derivatives, this solution helps you understand decreasing behavior and concavity at point A. Suitable for students in Grades 11-12.

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