**a.** The​ numbers, if​ any, at which f has a relative maximum. What are these relative​ maxima?


To determine where the function \( f(x) \) has a relative maximum and identify the values of these relative maxima, we can follow these steps:

### Steps to Identify Relative Maxima

1. **Find the derivative** \( f'(x) \) of the function \( f(x) \). The derivative represents the slope of the tangent line at any point on the curve.

2. **Set the derivative equal to zero** and solve for \( x \). The points where \( f'(x) = 0 \) are called critical points. These points are potential locations for relative maxima or minima.

3. **Use the second derivative test** or analyze the sign changes of the first derivative \( f'(x) \) around the critical points to determine whether each critical point is a relative maximum, relative minimum, or neither:
   - If \( f''(x) > 0 \) at a critical point, the function has a relative minimum there.
   - If \( f''(x) < 0 \) at a critical point, the function has a relative maximum there.
   - If \( f''(x) = 0 \), the test is inconclusive, and further analysis is needed.

4. **Evaluate the function** \( f(x) \) at the critical points where a relative maximum is found to identify the actual maximum value.

### Analysis Based on the Graph

Looking at the graph:
- Identify points where the function changes from increasing to decreasing. These points often correspond to relative maxima.
- These points should have a horizontal tangent (where the derivative is zero).

### Conclusion

Without the specific function or the exact coordinates from the graph, here is how you would interpret relative maxima from the graph:

1. **Identify critical points**: Look at the graph to see where the slope of the tangent is zero. These are your candidates for maxima.

2. **Check for relative maxima**: At each critical point, observe if the function value increases before and decreases after. This indicates a relative maximum.

3. **List the maxima**: Once identified, write down the \( x \)-values (and corresponding \( y \)-values) of these relative maxima.

If you can provide more details from the graph or function, I can help pinpoint these values specifically.

Would you like to provide any more details or continue with another part of the problem?

### Follow-up Questions

1. Do you need help finding the first derivative \( f'(x) \) for a specific function?
2. Would you like an example on how to perform the second derivative test?
3. Can you identify the exact coordinates of the critical points from the graph?
4. Are you interested in understanding the graphical interpretation of relative maxima in more detail?
5. Do you need assistance in understanding how to evaluate \( f(x) \) at the critical points?

### Tip

Relative maxima occur at points where the function changes from increasing to decreasing, and their identification often requires analyzing the derivative's sign changes around critical points.

Learn how to identify relative maxima of a function using calculus. Explore the steps involving derivatives and the second derivative test to pinpoint where the function reaches its highest points. Suitable for advanced high school students.

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