https://file.aiquickdraw.com/mathbot/file/1729475981696rhqtqwcf.jpg

The graph in the image shows a continuous curve with a few critical points where the slope changes direction, indicating extrema or stationary points.

### To locate and classify the extrema:
1. **Relative Maximum** occurs when the function changes from increasing to decreasing, forming a peak.
2. **Relative Minimum** occurs when the function changes from decreasing to increasing, forming a trough.
3. **Stationary Points** are where the derivative (slope) is zero, which can be at maxima, minima, or flat points.

### Observing the graph:

- **Relative Maximum at x = -6**: The function increases, reaches a peak around \(x = -6\), with a \(y\)-value of approximately 6, then decreases. Hence, this is a **relative maximum**.
  
- **Relative Minimum at x = 2**: The function decreases, reaches a trough around \(x = 2\), with a \(y\)-value of approximately -5, then increases. Hence, this is a **relative minimum**.

- **Relative Maximum at x = 7**: The function increases again and peaks around \(x = 7\), with a \(y\)-value of approximately 3. Hence, this is another **relative maximum**.

### Extrema Summary (from smallest to largest \(x\)):

1. **Relative Maximum** at \(x = -6\), \(y = 6\)
2. **Relative Minimum** at \(x = 2\), \(y = -5\)
3. **Relative Maximum** at \(x = 7\), \(y = 3\)

### Stationary points:
The function has stationary points at the relative extrema identified above. There are no other notable stationary points where the curve flattens without being an extremum.

Would you like further details or have any questions?

### Related questions:
1. How do we mathematically find the extrema of a function using calculus?
2. What is the significance of relative maxima and minima in real-world applications?
3. How can we identify whether a stationary point is a maximum or minimum without a graph?
4. What are inflection points, and how can they be identified?
5. Can a function have more than one absolute maximum or minimum?

**Tip:** When finding extrema, always check the derivative to locate critical points and determine if they correspond to maxima or minima by using the second derivative test or analyzing the slope.

Locate and classify all extrema in the graph. By classifying the extrema, we mean listing whether each extremum is a relative or absolute maximum or minimum. Also, locate any stationary points or singular points that are not relative extrema.

Learn how to locate and classify extrema (relative maxima and minima) in a function graph. This example covers identifying stationary points and applying key calculus concepts such as the first and second derivative tests. Suitable for students in Grades 11-12.

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