https://file.aiquickdraw.com/mathbot/file/1726366082090v3iqhkj2.jpg

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(

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?

This problem involves analyzing the local and global maxima of a polynomial function with specific values at points like f(2), f(3), f(4), and f(6). Learn how to use derivative tests to determine the behavior of the function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation