https://file.aiquickdraw.com/mathbot/file/1727381157181uc6blq09.jpg

Let's analyze the graph based on the questions:

### 1. At which labeled point is \( f(x) \) greatest?
To determine where \( f(x) \) is greatest, we need to identify the highest point on the graph. From the image, point **G** appears to be the highest point, so:
- \( f(x) \) is greatest at **G**.

### 2. At which labeled point is \( f(x) \) least?
The lowest point on the graph will give the smallest value of \( f(x) \). From the image, point **F** is the lowest, so:
- \( f(x) \) is least at **F**.

### 3. At which labeled point is \( f'(x) \) greatest?
\( f'(x) \), the derivative of \( f(x) \), represents the slope of the graph. We are looking for the steepest positive slope. This occurs where the curve is rising the fastest, which appears to be around point **C**.

### 4. At which labeled point is \( f'(x) \) least?
Here, we are looking for the steepest negative slope, where the graph is falling the fastest. This happens around point **B**, where the graph is steeply decreasing.

Let me know if you would like more details or have any further questions.

### Related Questions:
1. How can we interpret the behavior of the graph near point E?
2. What are the local maxima and minima of the function based on this graph?
3. How does the slope of the graph change between points B and C?
4. Can the concavity of the function be determined from this graph?
5. What does the second derivative \( f''(x) \) tell us about the graph near points A and D?

### Tip:
To better understand the function's behavior, focus on how the slope (the derivative) changes between points, which will help identify critical points and inflection points.

Use the graph of y = f(x) in the figure to answer the following questions:
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 graph of a function f(x) and its first derivative f'(x) to identify points of maximum and minimum. This example explores finding critical points, including where f(x) is greatest or least, and where f'(x) shows the greatest and least slopes.

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