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To solve the problem, we need to determine the interval where the graph of the function \( f \) is concave down based on the table that describes the rate of change of \( f \).

### Information from the table:
- **Interval \( 0 < x < 1 \):** Rate of change is increasing.
- **Interval \( 1 < x < 2 \):** Rate of change is positive and constant.
- **Interval \( 2 < x < 3 \):** Rate of change is decreasing.
- **Interval \( 3 < x < 4 \):** Rate of change is negative and constant.

### Key Concept:
- A function is **concave down** when its rate of change is decreasing (i.e., its second derivative is negative or the slope of the tangent line is decreasing).
- Therefore, concavity depends on how the rate of change behaves: when the rate of change (or first derivative) is decreasing, the function is concave down.

### Solution:
The table shows that in the interval \( 2 < x < 3 \), the rate of change is **decreasing**. This indicates that the function is concave down on this interval.

### Answer:
The correct interval where the graph of \( f \) is concave down is **C. \( 2 < x < 3 \)**.

Would you like further details or have any questions?

Here are some related questions:
1. What is the relationship between the second derivative and concavity?
2. How does the sign of the second derivative affect the shape of a function's graph?
3. What happens to concavity when the rate of change is constant?
4. Can a function be concave up if its rate of change is positive but decreasing?
5. How would we graph a function based on the information about its rate of change?

**Tip:** Always check the behavior of the first and second derivatives to determine the concavity of a function.

On which of the following intervals is the graph of f concave down? (Given a table with rates of change for different intervals.)

Learn how to identify concave down intervals based on the rate of change of a function. This problem explains key concepts of concavity and derivatives, using data from a table for different x intervals. Suitable for students in Grades 10-12.

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