The given problem provides specific values for a polynomial function $f(x)$ at certain points:

• $f(2) = 0$
• $f(3) = 6$
• $f(4) = 0$
• $f(6) = -1$

The question asks which of the following must be true, given this information:

Analysis:

General Properties of Polynomials:

• Polynomials are continuous and differentiable functions, meaning we can apply concepts like local and global extrema.
• A local maximum occurs where the derivative changes from positive to negative, indicating a peak.
• A global maximum refers to the highest value of the function over its entire domain.

Analyzing the Given Values:

• At $x = 2$ and $x = 4$, the function equals zero. This suggests the possibility of roots or turning points.
• $f(3) = 6$ is higher than the function values at $x = 2$ and $x = 4$, so $x = 3$ could be a local maximum, because the function increases to 6 and then decreases back to 0.

Examining the Choices:

• (A) $f$ has a local maximum when $x = 3$.

  • This is likely true because $f(x)$ reaches a peak at $x = 3$ and then decreases afterward.

• (B) $f$ has a global maximum when $x = 3$.

  • This might also be true because 6 is the highest value given for $f(x)$, and no larger values are suggested.

• (C) $f$ has a local maximum for some $x$ with $2 < x < 4$.

  • This is true because $x = 3$ is in the interval $2 < x < 4$, and we established that ( f(