Let's analyze the graph to determine the absolute maximum and minimum values, as well as any local extrema.

Analysis:

• Points of Interest:
  • $(-2, -2)$: Closed circle, meaning the function is defined at this point.
  • $(-1, -4)$: Open circle, meaning the function is not defined at this point (potential hole in the graph).
  • $(3, 0)$: Closed circle, meaning the function is defined at this point.
  • $(4, 2)$: Open circle, meaning the function is not defined at this point (potential hole in the graph).
  • $(6, 1)$: Closed circle, meaning the function is defined at this point.

Absolute Maximum:

• The highest point on the graph is at $(4, 2)$, but since it is an open circle, it means the function is not defined there. Therefore, this point cannot be an absolute maximum.
• The next highest point where the function is defined is $(6, 1)$, but this is not the highest point on the graph, so there is no absolute maximum.

Conclusion: There is no absolute maximum for $y = f(x)$.

Absolute Minimum:

• The lowest point where the function is defined is $(-2, -2)$, but this is not the lowest point on the graph.
• The graph has a lower point at $(-1, -4)$, but since this is an open circle, the function is not defined there.
• Hence, the absolute minimum is at $(-2, -2)$.

Conclusion: The absolute minimum for $y = f(x)$ is $-2$ at $x = -2$.

Local Maxima and Minima:

• Local Minimum: The point $(-2, -2)$ is a local minimum since the graph dips down to this point before rising again.
• Local Maximum: The point $(4, 2)$ is a local maximum, but it's an open circle, so it's not defined in the function. Thus, the only other point to consider is $(6, 1)$, but since it's at the edge and not followed by a decrease, it isn't a local maximum either.

Final Answers:

• Part 1 (Absolute Maximum):

  • B. There is no absolute maximum for $y = f(x)$.

• Part 2 (Absolute Minimum):

  • A. The absolute minimum is $-2$ at $x = -2$.

Would you like to go over anything in more detail?

Here are 5 related questions you might find interesting:

1. How can you distinguish between a local maximum and an absolute maximum on a graph?
2. What conditions must be met for a point to be considered an absolute minimum?
3. How do open and closed circles affect the interpretation of a graph?
4. Can a function have more than one absolute maximum or minimum?
5. How do you determine the intervals of increase and decrease from a graph?

Tip: Always check if a point is included (closed circle) or not included (open circle) when identifying absolute or local extrema on a graph. This ensures accuracy in your analysis.