https://file.aiquickdraw.com/mathbot/file/172705133701416w790au.jpg

The graph in the image shows three key points:

1. \( (2, 7) \)
2. \( (4, 3) \)
3. \( (5, 4) \)
4. \( (6, 2) \)

We need to find the absolute maximum and minimum values for the function \( y = f(x) \), as well as any local extrema if they exist.

- **Absolute Maximum**: The highest point on the graph is at \( (2, 7) \), so the absolute maximum value is 7.
- **Absolute Minimum**: The lowest point on the graph is at \( (6, 2) \), so the absolute minimum value is 2.

- **Local Maxima and Minima**:
   - The local maximum occurs at \( (2, 7) \) and \( (5, 4) \), as these points are higher than the surrounding points.
   - The local minimum occurs at \( (4, 3) \), as this point is lower than its immediate surroundings.

Therefore, the correct answer is to fill in:
- Absolute maximum: \( 7 \) (at \( x = 2 \)).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the definition of a local maximum in calculus?
2. How do you find absolute extrema from a graph?
3. What is the difference between absolute and local extrema?
4. Can there be more than one absolute maximum in a function?
5. How does the first derivative help in identifying extrema?

**Tip**: Always check both endpoints and critical points when determining absolute maxima and minima.

For the graph of a function y = f(x), find the absolute maximum and minimum, and identify any local maxima or minima.

Learn how to find the absolute maximum and minimum of a function based on its graph, and identify local maxima and minima. Example includes key points like (2,7), (4,3), and (6,2).

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