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Let's analyze the given information and the choices:

We are given a polynomial function \( f(x) \), and several specific values for the function at certain points:
- \( f(2) = 0 \),
- \( f(3) = 6 \),
- \( f(4) = 0 \),
- \( f(6) = -1 \).

Now, let's interpret each option:
- **(A)** \( f \) has a local maximum when \( x = 3 \).
- **(B)** \( f \) has a global maximum when \( x = 3 \).
- **(C)** \( f \) has a local maximum for some \( x \) with \( 2 < x < 4 \).
- **(D)** \( f \) has a global maximum for some \( x \) with \( 2 < x < 4 \).

### Observations:
1. The polynomial passes through \( f(2) = 0 \) and \( f(4) = 0 \), meaning that \( x = 2 \) and \( x = 4 \) are roots of the polynomial.
2. Between these two roots, at \( x = 3 \), the function reaches a value of \( f(3) = 6 \), which is higher than the values at both \( x = 2 \) and \( x = 4 \).
   - Therefore, it's very likely that there is a **local maximum at \( x = 3 \)** because the function reaches a peak there between two lower values.
3. The function decreases after \( x = 4 \), as \( f(6) = -1 \), which is lower than the values around \( x = 3 \).

### Conclusion:
- Option **(A)** is true, as \( f \) has a local maximum at \( x = 3 \).
- Option **(B)** is not necessarily true because although \( f(3) = 6 \) might be a high value, it is not guaranteed to be the global maximum without further information about the polynomial beyond \( x = 6 \).
- Option **(C)** is essentially saying the same thing as option **(A)** but in a different way, so this is also true.
- Option **(D)** is false because the function decreases after \( x = 4 \), meaning \( x = 3 \) is likely the highest point within \( 2 < x < 4 \), but not necessarily a global maximum.

Thus, the correct answers are **(A)** and **(C)**.

Let me know if you want further clarification! 

Here are 5 related questions to consider:
1. What conditions guarantee that a polynomial has a local maximum at a given point?
2. Can a polynomial have more than one global maximum?
3. How do the first and second derivative tests help identify local maxima and minima?
4. How can we determine if a critical point is a local minimum or a maximum?
5. What are the properties of polynomials regarding their end behavior?

**Tip:** When analyzing polynomials, the behavior between roots is critical for understanding maxima and minima points.

Suppose f(x) is a polynomial function. If f(2) = 0, f(3) = 6, f(4) = 0, and f(6) = -1, which of the following must be true? A) f has a local maximum when x = 3. B) f has a global maximum when x = 3. C) f has a local maximum for some x with 2 < x < 4. D) f has a global maximum for some x with 2 < x < 4.

This problem explores local and global maxima for a polynomial function given specific values at f(2), f(3), f(4), and f(6). The analysis involves understanding critical points and polynomial behavior between roots. Suitable for advanced high school students in Grades 11-12.

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