Let's analyze the graph to identify the absolute and local maximum and minimum values of the function $y = f(x)$.

1. Absolute Maximum Value:

The absolute maximum is the highest point on the graph. The highest point visible on this graph occurs at $y = 2$ (the point is at $x = 4$, $y = 2$).

• Answer: 2

2. Absolute Minimum Value:

The absolute minimum is the lowest point on the graph. The lowest point visible on this graph occurs at $y = -1$ (the point is at $x = 0$, $y = -1$).

• Answer: -1

3. Local Maximum Values:

Local maximum values occur at the peaks of the graph (high points between intervals where the graph goes up and down). There are two such peaks:

• One at $x = 1$, $y = 1$

• Another at $x = 4$, $y = 2$ (which is also the absolute maximum).

• Answer: 1, 2

4. Local Minimum Values:

Local minimum values occur at the valleys of the graph (low points between intervals). There are two such valleys:

• One at $x = 0$, $y = -1$ (which is also the absolute minimum).

• Another at $x = 3$, $y = 0$.

• Answer: -1, 0

Summary of Correct Answers:

1. Absolute Maximum Value: 2
2. Absolute Minimum Value: -1
3. Local Maximum Value(s): 1, 2
4. Local Minimum Value(s): -1, 0

Would you like further clarification on any of these concepts or steps?

Here are 5 additional questions to deepen understanding:

1. How is a local maximum different from an absolute maximum?
2. Can a function have multiple absolute maximum values?
3. Why is it important to distinguish between local and absolute extrema?
4. How does the behavior of the function at critical points help us identify local maxima and minima?
5. How would this analysis change if we had an open interval instead of a closed one?

Tip: To confirm local extrema, you can also use the first and second derivative tests when dealing with a function's formula!