From the description of the graph, we can analyze the given function's absolute and local maximum and minimum values. Here's a breakdown based on your description:

Absolute Maximum Value:

• The absolute maximum occurs at the highest point the function reaches over the entire graph.
• The function reaches a value of 5 at $(4, 5)$, which is the highest value it attains.

Absolute Maximum Value = 5

Absolute Minimum Value:

• The absolute minimum occurs at the lowest point the function reaches over the entire graph.
• The function reaches a value of 1 at $(7, 1)$, which is the lowest value it attains.

Absolute Minimum Value = 1

Local Maximum Values:

• Local maxima occur at points where the function changes from increasing to decreasing.
• From your description, the local maxima are:
  • At $(1, 4)$, where the function increases to a peak and then decreases.
  • At $(4, 5)$, where the function reaches a peak before decreasing.

Local Maximum Values = 4, 5

Local Minimum Values:

• Local minima occur at points where the function changes from decreasing to increasing.
• From your description, the local minima are:
  • At $(2, 2)$, where the function decreases and then increases.
  • At $(5, 3)$, where the function decreases and then sharply increases.

Local Minimum Values = 2, 3

Final Answers:

• Absolute Maximum Value = 5
• Absolute Minimum Value = 1
• Local Maximum Values = 4, 5
• Local Minimum Values = 2, 3

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Let me know if you need any further clarifications or explanations!

Related Questions:

1. What is the difference between local and absolute extrema?
2. How can we identify inflection points on a graph?
3. What is the role of the first derivative in finding maximum and minimum points?
4. Can there be multiple local maxima or minima in a function?
5. How do you graph a piecewise function with multiple curves?

Tip: When identifying maxima and minima, it's useful to look at points where the function changes direction—either from increasing to decreasing (local maxima) or from decreasing to increasing (local minima).