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Let's analyze the graph and questions.

We are given a graph with points \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \) labeled, along with the function \( y = f(x) \). The task involves analyzing the slope and behavior of the function at these points.

### Here's a breakdown of each question:

1. **At which labeled point(s) is the slope of the curve positive?**
   - A positive slope means the curve is increasing. From the graph, we see the curve is increasing between points \( D \) and \( B \). So, the slope at points between these two is positive, but none of these points are specifically labeled. This could mean point **B** is likely where the slope is still positive.

2. **At which labeled point(s) is the slope of the curve negative?**
   - A negative slope means the curve is decreasing. From the graph, we see the curve is decreasing from points \( A \) to \( B \), as well as after \( D \) towards \( F \). Therefore, the slope is negative at **A**, **C**, and **F**.

3. **At which labeled point(s) is the slope of the curve zero?**
   - A slope of zero means the function is neither increasing nor decreasing (a horizontal tangent). Looking at the graph, it is flat at points **B**, **D**, and **E**, indicating a zero slope.

4. **Where is the function increasing?**
   - The function increases where the slope is positive. As observed, it increases between **D** and **B**.

5. **Where is the function decreasing?**
   - The function decreases where the slope is negative. This occurs from **A** to **B** and after **D** towards **F**.

6. **Which labeled point has the greatest slope?**
   - The greatest positive slope occurs between points **D** and **B**, so point **B** likely has the steepest positive slope.

7. **Which labeled point has the least slope?**
   - The least slope (most negative or steepest downward slope) occurs at point **C** since the curve steeply declines here.

### Summary of answers:
1. **Positive slope**: Point \( B \)
2. **Negative slope**: Points \( A \), \( C \), \( F \)
3. **Zero slope**: Points \( B \), \( D \), \( E \)
4. **Increasing**: From \( D \) to \( B \)
5. **Decreasing**: From \( A \) to \( C \), and after \( D \) to \( F \)
6. **Greatest slope**: Point \( B \)
7. **Least slope**: Point \( C \)

Do you have any questions or want further details on any part of this?

### 5 Follow-up Questions:
1. What is the significance of a zero slope in terms of turning points?
2. How do you determine concavity from a graph?
3. What would be the second derivative at points with a zero slope?
4. How can you estimate the exact values of slopes at points based on the graph?
5. What are the conditions for a local maximum or minimum at a given point?

### Tip:
When analyzing a graph, remember that the slope gives you the rate of change, while the second derivative (if available) tells you how the slope is changing.

At which labeled points is the slope of the curve positive, negative, or zero? Where is the function increasing or decreasing, and which point has the greatest and least slope?

This problem involves determining the slope of a curve at different labeled points, identifying where the function is increasing or decreasing, and finding the points with the greatest and least slopes. Suitable for students in Grades 9-11.

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