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The table shows the average rates of change for a polynomial function \( f(x) \) over different intervals. Let's analyze the situation step by step to evaluate which statement is true about the graph.

The key information from the table:
- Interval \([1, 2]\): Rate of change = \(-1.5\)
- Interval \([2, 3]\): Rate of change = \(17.5\)
- Interval \([3, 4]\): Rate of change = \(50.5\)
- Interval \([4, 5]\): Rate of change = \(98.5\)

The rate of change of a function represents its slope over a particular interval. If the rate of change is increasing, the graph of the function is concave up. If the rate of change is decreasing, the graph is concave down.

### Analysis:
- From \([1, 2]\) to \([2, 3]\), the rate of change increases from \(-1.5\) to \(17.5\).
- From \([2, 3]\) to \([3, 4]\), the rate of change further increases from \(17.5\) to \(50.5\).
- From \([3, 4]\) to \([4, 5]\), the rate of change again increases from \(50.5\) to \(98.5\).

This consistent increase in the average rate of change suggests that the graph of \( f \) is concave **up**, not concave down, over the intervals being discussed.

### Evaluation of Statements:
1. **"The graph is concave up on the interval [1, 3]."**  
   - **True**. The rate of change is increasing on this interval, indicating concavity up.
   
2. **"The graph is concave down on the interval [1, 3]."**  
   - **False**. The graph is concave up because the rate of change is increasing between these points.
   
3. **"The graph is decreasing on the interval [1, 3]."**  
   - **False**. The rate of change is negative at \([1, 2]\) but becomes positive by \([2, 3]\), so the graph is not decreasing over the entire interval.

4. **"There is an inflection point at \( x = 2 \)."**  
   - **Uncertain** based on the given data. The rate of change switches from negative to positive around \(x = 2\), suggesting that there could be an inflection point, but further information would be required to confirm this.

The best choice among the options given is that the graph is **concave up on the interval [1, 3]**, but you marked the second statement, which is incorrect.

### Would you like further clarification, or do you have any other questions?

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Here are 5 related questions for further exploration:
1. What does it mean for a function to be concave up versus concave down?
2. How can you determine whether a function has an inflection point based on rates of change?
3. What is the relationship between the second derivative of a function and its concavity?
4. Can a polynomial function have multiple inflection points?
5. How does the sign of the rate of change relate to increasing and decreasing intervals?

**Tip**: To confirm inflection points, check where the second derivative changes sign.

Which of the following statements is true about f? Based on the average rate of change of a polynomial over various intervals.

Analyze the concavity of a polynomial function based on the average rate of change across intervals. This problem demonstrates how changes in the average rate indicate whether a function is concave up or down. Suitable for students in Grades 11-12.

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