https://file.aiquickdraw.com/mathbot/file/1727051022039nnyk7hhu.jpg

In the image, you're asked to find the **absolute maximum** and **absolute minimum** of a function \( y = f(x) \) from its graph, along with any local maxima or minima. The graph appears to show several points of interest.

### Steps for identifying extrema:

1. **Identify points on the graph**: The function graph shows the following key points:
   - \( (2, 7) \)
   - \( (5, 4) \)
   - \( (6.5, 2) \)
   - \( (4, 3) \)
   - \( (4.3, 1) \)

2. **Absolute Maximum**:
   - The absolute maximum is the highest point on the graph.
   - From the provided graph, the highest \( y \)-value occurs at \( (2, 7) \). So, the **absolute maximum** is 7 when \( x = 2 \).

3. **Absolute Minimum**:
   - The absolute minimum is the lowest point on the graph.
   - The lowest point seems to be \( (6.5, 2) \), meaning the **absolute minimum** is 2 when \( x = 6.5 \).

4. **Local maxima and minima**:
   - Local maximum: The function decreases after \( (2, 7) \), so this point is a local maximum.
   - Local minimum: The function increases after \( (6.5, 2) \), so this point is a local minimum.

### Conclusion:
- **Absolute Maximum** is \( 7 \) at \( x = 2 \).
- **Absolute Minimum** is \( 2 \) at \( x = 6.5 \).

You should select option **A** and fill in the value 7 for the absolute maximum.

Would you like more details or have any questions?

### Related Questions:
1. How do you distinguish between local and absolute extrema?
2. What are critical points, and how do they relate to extrema?
3. How can you confirm if a point is a local maximum or minimum?
4. What is the difference between a continuous and discontinuous function in terms of extrema?
5. Can there be more than one absolute maximum or minimum on a graph?

**Tip**: Always look for turning points on the graph where the slope changes direction to identify local maxima or minima.

For the graph of a function y = f(x) shown to the right, find the absolute maximum and the absolute minimum, if they exist. Identify any local maxima or local minima.

Learn how to identify the absolute maximum and minimum of a function based on its graph. This problem demonstrates key concepts in graph analysis, including extrema and the application of the Extreme Value Theorem, suitable for high school students (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