The graph you've uploaded appears to be a function with several important points, and you are asking to identify the absolute maximum, absolute minimum, local maximum, and local minimum.

Key points from the graph:

1. $(-2, 2)$
2. $(0, 0)$
3. $(2, 4)$
4. $(4, 3)$
5. Dashed line extending vertically at $x = 5$

Step-by-step identification:

• Local maxima occur at peaks in the function where the function value is higher than at nearby points.
• Local minima occur at troughs in the function where the function value is lower than at nearby points.
• Absolute maxima is the highest point of the function over its entire domain.
• Absolute minima is the lowest point of the function over its entire domain.

Identifying Local and Absolute Extremes:

• Local maximum:

  • At $(2, 4)$, the function reaches its highest local point within a neighborhood, but it isn't necessarily the highest point overall.

• Local minimum:

  • At $(0, 0)$, the function dips to its lowest point in a small region.

• Absolute maximum:

  • The absolute maximum is the highest point on the graph. Here, $(2, 4)$ appears to be the highest point, so it would be the absolute maximum.

• Absolute minimum:

  • The absolute minimum is the lowest point on the graph. $(0, 0)$ seems to be the lowest point, which means it is the absolute minimum.

Summary:

• Absolute Maximum: $(2, 4)$
• Absolute Minimum: $(0, 0)$
• Local Maximum: $(2, 4)$
• Local Minimum: $(0, 0)$

Let me know if you need further details or have any questions!

Here are five related questions to explore further:

1. How can we find the exact points of maximum and minimum from the function's equation?
2. Can a function have more than one absolute maximum or minimum?
3. How do we determine if a point is a local extremum using calculus?
4. What role do derivative tests play in finding local maxima and minima?
5. How can concavity and inflection points affect the behavior of maxima and minima?

Tip: The first derivative test helps in determining if a point is a local maximum or minimum by analyzing the sign changes of the derivative at that point.