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.

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.