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For the function shown, at what labeled points is the slope of the graph positive? Negative? At which labeled point does the graph have the greatest (i.e., most positive) slope? The least slope (i.e., negative and with the largest magnitude)?

### Analyzing the Graph
The graph shows a smooth curve with points labeled \( A, B, C, D, E, \) and \( F \). To determine the slope at these points, we need to evaluate whether the tangent to the curve at each point is positive, negative, or zero.

### Slope Analysis
1. **Positive Slope:**
   The slope is positive where the function is increasing (moving upwards from left to right).

   - Points **C** and **D** have a positive slope.

2. **Negative Slope:**
   The slope is negative where the function is decreasing (moving downwards from left to right).

   - Points **A, B,** and **F** have a negative slope.

3. **Zero Slope:**
   The slope is zero at local maxima, minima, or points of horizontal tangency.

   - Points **B, D,** and **E** have zero slope (as these are the points where the curve changes direction).

### Greatest Positive Slope
The greatest positive slope corresponds to the point where the curve has the steepest ascent. This appears to be:

- **Point C**, as the curve is steeply increasing there.

### Least Slope (Most Negative)
The least slope (most negative) is at the point where the curve is steeply descending. This occurs at:

- **Point F**, as the graph is steeply decreasing at this point.

### Summary
- **Positive slope:** Points C and D.
- **Negative slope:** Points A, B, and F.
- **Greatest positive slope:** Point C.
- **Least (most negative) slope:** Point F.

Would you like to discuss any part of the graph or solution in more detail? 

### Follow-Up Questions
1. How would you identify points of inflection in this graph?
2. What is the significance of a zero slope in relation to local extrema?
3. How can we approximate the slopes at each labeled point more accurately?
4. Could the concavity of the curve change at any labeled points?
5. How can the graph be used to estimate the derivative values at each point?

### Tip
Always analyze the tangent lines to the curve to determine the sign and relative magnitude of slopes at specific points.

Learn how to analyze the slopes at different points on a function graph. This problem focuses on finding positive and negative slopes, identifying the point with the greatest positive slope, and determining the point with the steepest negative slope. Suitable for students in Grades 11-12 studying calculus.

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