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.