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The problem in your image involves analyzing a graph of a function \( f(x) \) and determining its behavior at labeled points. Here's how you would typically approach the four questions:

1. **At which labeled point is \( f(x) \) greatest?**
   - This is asking for the point on the graph where the function \( f(x) \) has its highest value. This would correspond to the highest point on the graph (maximum point).

2. **At which labeled point is \( f(x) \) least?**
   - This is asking for the point where \( f(x) \) is at its lowest value, or the minimum point on the graph.

3. **At which labeled point is \( f'(x) \) greatest?**
   - \( f'(x) \) represents the derivative of the function, which tells us the slope of the function at each point. The point where \( f'(x) \) is greatest is where the slope is steepest in the positive direction (i.e., the graph is increasing most rapidly).

4. **At which labeled point is \( f'(x) \) least?**
   - Here, you are looking for the point where the slope is steepest in the negative direction (i.e., where the graph is decreasing most rapidly).

### How to solve:
- **For questions (1) and (2)**, focus on the height of the graph to find the maximum and minimum values of \( f(x) \).
- **For questions (3) and (4)**, examine the slope of the graph. The steepest upward slope will correspond to the maximum \( f'(x) \), and the steepest downward slope will correspond to the minimum \( f'(x) \).

If you could zoom in on the graph and labeled points, it would be easier to give specific answers based on the actual graph.

Let me know if you want further clarification or if you'd like me to process specific parts of the graph!

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Here are 5 relative questions that build on the same concepts:

1. What is the relationship between the critical points of \( f(x) \) and the values of \( f'(x) \)?
2. How do you determine the concavity of a function based on the graph of \( f(x) \)?
3. What happens to the graph of \( f(x) \) at points where \( f'(x) = 0 \)?
4. Can a function have a maximum \( f(x) \) value but a zero derivative at that point? Why or why not?
5. How does the second derivative \( f''(x) \) affect the shape of the graph of \( f(x) \)?

**Tip:** When finding where \( f(x) \) is greatest or least, always focus on the highest and lowest vertical points of the graph!

The following problem is similar to a problem from your textbook. Use the graph of y = f(x) in the figure below to answer the questions that follow: (1) At which labeled point is f(x) greatest? (2) At which labeled point is f(x) least? (3) At which labeled point is f'(x) greatest? (4) At which labeled point is f'(x) least?

Learn how to analyze the behavior of a function f(x) and its derivative f'(x) using graph-based analysis. Determine points where f(x) is greatest or least and where the slope of the graph (f'(x)) is steepest, applying key calculus concepts such as maxima, minima, and critical points. Suitable for Grades 10-12 students studying calculus.

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