The table gives selected values of polynomial h(x).

x	0	2	4	6	8
h(x)	–8	0	4	0	18

Which statement is true if all zeros of h are given in the table?
The function h has a local minimum at (6, 0).
The function h has a global minimum at (0, –8).
The function h has a local maximum at (2, 0).
The function h has a global maximum at (8, 18).

From the given table, we have values of the polynomial \( h(x) \) at different points:

| \( x \)  | 0 | 2 | 4 | 6 | 8 |
| --- | --- | --- | --- | --- | --- |
| \( h(x) \) | -8 | 0 | 4 | 0 | 18 |

We need to analyze which statement is true based on this information. Here's how we can assess each statement:

### 1. **The function \( h \) has a local minimum at (6, 0).**
- At \( x = 6 \), \( h(6) = 0 \). To determine if this is a local minimum, we check if the values before and after \( x = 6 \) are larger. 
  - \( h(4) = 4 \) (greater than 0), and \( h(8) = 18 \) (also greater than 0).
  - This suggests that \( x = 6 \) is indeed a local minimum.

### 2. **The function \( h \) has a global minimum at (0, –8).**
- The function reaches \( h(0) = -8 \), which is the lowest value in the table. This indicates that \( (0, -8) \) is the global minimum.

### 3. **The function \( h \) has a local maximum at (2, 0).**
- At \( x = 2 \), \( h(2) = 0 \). To determine if this is a local maximum, we check the values around it.
  - \( h(0) = -8 \) (less than 0), and \( h(4) = 4 \) (greater than 0), which suggests that \( x = 2 \) could be a local maximum.
  
### 4. **The function \( h \) has a global maximum at (8, 18).**
- At \( x = 8 \), \( h(8) = 18 \) is the highest value in the table, making this point the global maximum.

### Conclusion:
- **The correct statement is:** "The function \( h \) has a global maximum at \( (8, 18) \)." This is because 18 is the highest value for \( h(x) \) in the table.

Do you need any further clarification or details?

---

### Related Questions:
1. What is the definition of a local maximum and minimum?
2. How do you determine global extrema for a function based on its values?
3. What additional information would be required to analyze the derivative of \( h(x) \)?
4. How would you graph this polynomial based on the given points?
5. How can a polynomial function's degree affect the number of local extrema?

### Tip:
Always check the values surrounding a point to verify if it's a local maximum or minimum.

The table gives selected values of polynomial h(x).
x    0    2    4    6    8
h(x)    –8    0    4    0    18
Which statement is true if all zeros of h are given in the table?
The function h has a local minimum at (6, 0).
The function h has a global minimum at (0, –8).
The function h has a local maximum at (2, 0).
The function h has a global maximum at (8, 18).

Analyze a polynomial function based on selected table values: x = {0, 2, 4, 6, 8} and h(x) = {-8, 0, 4, 0, 18}. Learn how to determine local minima, local maxima, global minima, and global maxima for polynomial functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation