Identify all of the global and local extremes of the graph 

To identify global and local extrema (maximums and minimums) of a function, we follow a systematic approach. Here is how to analyze the graph of a function for its extremes:

### General Steps to Find Extrema:
1. **Find the derivative** \( f'(x) \):
   - The derivative of a function helps us identify critical points where the slope of the tangent is zero (\( f'(x) = 0 \)) or where \( f'(x) \) does not exist.
   
2. **Identify critical points**:
   - Set \( f'(x) = 0 \) and solve for \( x \). These points are where local extrema can occur.
   - Also check points where \( f'(x) \) does not exist but \( f(x) \) does, as they might be critical points.

3. **Determine the nature of critical points**:
   - Use the **Second Derivative Test**:
     - If \( f''(x) > 0 \) at a critical point, it's a **local minimum**.
     - If \( f''(x) < 0 \) at a critical point, it's a **local maximum**.
     - If \( f''(x) = 0 \), the test is inconclusive, and you may need to use other methods like analyzing the behavior around the point.
   - Alternatively, use the **First Derivative Test** to determine if a critical point is a maximum, minimum, or neither:
     - If \( f'(x) \) changes from positive to negative at a critical point, it is a **local maximum**.
     - If \( f'(x) \) changes from negative to positive at a critical point, it is a **local minimum**.

4. **Check endpoints (if the domain is restricted)**:
   - If the function is defined on a closed interval \([a, b]\), evaluate \( f(a) \) and \( f(b) \) as potential extrema.

5. **Identify global extrema**:
   - Compare the values of \( f \) at all critical points and endpoints to find the **global maximum** and **global minimum** over the given domain.

If you provide a specific function or graph, we can go through these steps together to find the local and global extrema. Alternatively, you can upload a graph or image for a visual analysis.

Would you like to continue with a specific example? 

### Follow-up questions:
1. What function would you like me to analyze for local and global extrema?
2. Do you need help with finding the derivative of a given function?
3. Would you like to see an example of the First Derivative Test in action?
4. Are you interested in understanding how to analyze a graph visually for extrema without calculus?
5. Would you like an explanation of how the Second Derivative Test works?

### Tip:
When identifying extrema, always check both critical points and endpoints if the domain is restricted; sometimes the highest or lowest values occur at the boundaries.

Identify all of the global and local extremes of the graph

Learn how to identify global and local extrema of a graph using calculus. This guide walks through the critical points, derivative tests, and how to compare values to find extrema. Suitable for students in Grades 11-12 and early college levels.

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