https://file.aiquickdraw.com/mathbot/file/1727186533079gh5lxrjs.jpg

Let's analyze the image and questions related to maxima and minima. Based on the curve in the diagram and the labeled points, we can categorize them as absolute maxima, absolute minima, relative maxima, relative minima, or neither.

### 1. Absolute Maximum:
An **absolute maximum** is the highest point on the entire graph. From the image:
- **Point B** is the highest point in the entire range of the function.

So, **B** is the **absolute maximum**.

### 2. Absolute Minimum:
An **absolute minimum** is the lowest point on the entire graph. From the image:
- **Point D** appears to be the lowest point on the graph.

So, **D** is the **absolute minimum**.

### 3. Relative Maximum:
A **relative maximum** is a point where the function reaches a local peak, meaning it's higher than the points immediately surrounding it, but not necessarily the highest overall. From the image:
- **Point B** is a local peak, so it's also a **relative maximum**.
- **Point F** is another local peak.

So, **B** and **F** are **relative maxima**.

### 4. Relative Minimum:
A **relative minimum** is a point where the function reaches a local low. It's lower than the points immediately surrounding it. From the image:
- **Point C** is a local minimum.
- **Point D** is a local minimum as well.

So, **C** and **D** are **relative minima**.

### 5. Neither Maximum nor Minimum:
Some points are neither maxima nor minima; they are inflection points or points where the function is neither at a peak nor a trough. From the image:
- **Point A** is not a maximum or minimum.
- **Point G** is not a maximum or minimum.
- **Point E** seems to be at a horizontal inflection point, so it's also not a maximum or minimum.

So, **A**, **G**, and **E** are **neither maxima nor minima**.

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### Answers:

- **Absolute Maxima:** B
- **Absolute Minima:** D
- **Relative Maxima:** B, F
- **Relative Minima:** C, D
- **Neither Maximum nor Minimum:** A, E, G

Would you like any more detailed explanations? Here are some related questions:

1. What is the difference between absolute and relative extrema?
2. Can a point be both a relative maximum and an absolute maximum?
3. How are inflection points different from extrema?
4. What happens at a point of inflection in terms of concavity?
5. Why do local extrema occur where the derivative is zero or undefined?

**Tip:** The first derivative test can help identify relative maxima and minima by analyzing the sign changes of the derivative around critical points.

Decide whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum, or neither. (Referring to the image with labeled points A, B, C, D, E, F, and G)

Learn how to identify absolute and relative maxima and minima from a graph. This problem covers key calculus concepts, including the use of critical points and the first derivative test. Suitable for high school students in Grades 10-12.

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