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**Identify all of the global and local extrema of the graph.**

To identify the global and local extrema of the graph, let's analyze the given curve:

1. **Local Minimum:** 
   - The graph reaches its lowest point around \((1, 0)\). This is the **local minimum** because it is the lowest point in a particular interval.
   
2. **Local Maximum:** 
   - The graph reaches a peak at two points: one around \((-4, 8)\) and the other around \((7, 19)\). These are **local maxima** because they are the highest points in their respective intervals.

3. **Global Minimum:** 
   - The global minimum of the graph is the same as the local minimum at \((1, 0)\), as it is the lowest point on the entire graph.

4. **Global Maximum:** 
   - The global maximum is at \((7, 19)\) since this is the highest point on the graph.

In summary:
- **Global Minimum:** \((1, 0)\)
- **Global Maximum:** \((7, 19)\)
- **Local Maximum:** \((-4, 8)\), \((7, 19)\)
- **Local Minimum:** \((1, 0)\)

Would you like more details or have any other questions? 

Here are 8 related questions for further exploration:

1. How do you distinguish between a global and local extremum?
2. What are the necessary conditions for a point to be a local extremum?
3. Can a function have more than one global maximum or minimum? Why or why not?
4. How do the first and second derivative tests help identify extrema?
5. What role does concavity play in determining the type of extremum?
6. How can you find extrema using calculus for a given function?
7. What are the differences between absolute and relative extrema?
8. How does the behavior of a function at infinity affect the global extrema?

**Tip:** When analyzing extrema on a graph, always check the endpoints and critical points where the derivative is zero or undefined.

Explore how to identify global and local extrema in graphs with detailed explanations and examples. Learn about the Extreme Value Theorem, first and second derivative tests, and more. Suitable for college-level students and anyone interested in calculus and graph analysis.

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