Based on the description of the graph with points labeled $a$, $b$, $c$, $d$, $r$, and $s$, we can analyze each point for whether it represents an absolute or local maximum, minimum, or neither.

1. Absolute Maximum: This is the highest point on the entire graph.
2. Absolute Minimum: This is the lowest point on the entire graph.
3. Local Maximum: A peak within a small region where the function's value is higher than nearby points but may not be the highest on the entire graph.
4. Local Minimum: A valley within a small region where the function's value is lower than nearby points but may not be the lowest on the entire graph.
5. Neither Maximum nor Minimum: A point that doesn't represent a peak or valley in the graph.

Let's break down the situation at each labeled point:

At $a$:

• The graph begins at a closed point above $a$ and is at its lowest point.
• Therefore, $a$ represents an absolute minimum.

At $b$:

• The graph has a closed point above $b$ (ends) and then starts again at an open point directly below that point.
• Since the point is open after the gap, it cannot be a maximum or minimum.
• Therefore, $b$ is neither a maximum nor a minimum.

At $c$:

• The graph begins at a closed point above $c$ and continues, but the graph does not indicate a peak or valley at this point.
• Therefore, $c$ is neither a maximum nor a minimum.

At $d$:

• The graph changes direction at a point above $d$, indicating a local minimum since the graph has a valley here.
• Therefore, $d$ represents a local minimum.

At $r$:

• The graph reaches a highest point above $r$, and then sharply changes direction.
• This point is a peak, so $r$ represents a local maximum.

At $s$:

• The graph ends at a closed point above $s$ but is not the highest point, nor is it at the lowest point.
• Therefore, $s$ is neither a maximum nor a minimum.

Final Answers:

• Absolute maximum: None.
• Absolute minimum: $a$.
• Local maximum: $r$.
• Local minimum: $d$.
• Neither a maximum nor a minimum: $b$, $c$, $s$.

Would you like further details or have any questions?

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Here are five related questions to explore:

1. What is the difference between an absolute and a local maximum?
2. Why can't an open point be considered a maximum or minimum?
3. How would the graph change if the point at $b$ were closed instead of open?
4. Can a function have more than one absolute maximum or minimum?
5. How do we determine if a critical point is a local minimum or maximum?

Tip: Always check whether a point is open or closed when determining maximum or minimum values on a graph.