To determine where $f(x)$ is decreasing and concave down on the given graph:

1. Decreasing: $f(x)$ is decreasing where the slope of the tangent to the curve (i.e., $f'(x)$) is negative.
2. Concave Down: $f(x)$ is concave down where the curvature of the graph bends downwards, or $f''(x) < 0$.

Analysis:

• Point A: The curve is increasing to the left of this point and starts decreasing to the right. Thus, $f(x)$ is decreasing near $A$, and the curve is bending downwards here, meaning $A$ satisfies the condition of being decreasing and concave down.
• Point B: The curve is still increasing, though concave down, so $B$ does not satisfy the condition.
• Point C: The curve is increasing and concave up at this point.
• Point D: The curve is decreasing but concave up, so it does not satisfy the condition.

Conclusion:

Point A is the only point where $f(x)$ is decreasing and concave down.

Would you like further clarification or a detailed explanation of the properties of concavity and monotonicity?

Here are some related questions to explore:

1. How do first and second derivatives influence the shape of a graph?
2. What is the relationship between concavity and inflection points?
3. How can you find where a function is increasing or decreasing algebraically?
4. What are examples of functions with similar properties to this graph?
5. How can you determine concavity using a graphing calculator or software?

Tip: Always use the second derivative test to verify concavity when analyzing functions!