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."

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.