https://file.aiquickdraw.com/mathbot/file/1727181849845p9iuuk59.jpg

Let's analyze the graph to determine the critical points and phenomena at those points.

### Critical Points:
The graph shows three important points where the slope is zero, which are the critical numbers. From the graph:

1. **At \( x = -2 \)**: The function reaches a local (and global) maximum at this point, as the curve peaks.
2. **At \( x = 1 \)**: The function reaches a local minimum at this point since the curve dips.
3. **At \( x = 3 \)**: There is a point where the function has a horizontal tangent, which suggests another critical point. However, it doesn't seem to be a maximum or minimum but rather a point of inflection or a flat spot.

### Correct Answer for Critical Numbers:
- The critical numbers are \( x = -2, 1, 3 \) based on where the derivative is zero or undefined.

### Classification of the Critical Numbers:
- **Relative maximum**: Occurs at \( x = -2 \).
- **Relative minimum**: Occurs at \( x = 1 \).
- **Absolute maximum**: The function reaches its highest point at \( x = -2 \), so this is also the absolute maximum.
- **Absolute minimum**: The lowest point is at \( x = 1 \), so this is also the absolute minimum.

The missing point is \( x = 3 \), which you should add to the list of critical numbers. It is neither a maximum nor a minimum, so it does not apply to those categories.

Would you like more details on how to classify critical points or further clarification?

### Relative Questions:
1. What is the difference between relative and absolute extrema?
2. How do we find critical points algebraically for a given function?
3. Why is \( x = 3 \) considered a critical point but not a maximum or minimum?
4. What is the significance of the derivative being zero at critical points?
5. How does the second derivative test help in determining the nature of critical points?

**Tip**: Always check the endpoints of a graph or interval when looking for absolute extrema.

Approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

This problem involves approximating critical points of a function from its graph and classifying them as relative maxima, minima, or absolute extrema. Learn how to identify critical numbers using the first and second derivative tests, suitable for students studying Calculus I.

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