Use the graph of a function 

yequals=​f(x) 

to find the absolute maximum and the absolute​ minimum, if they exist. Identify any local maximum values or local minimum values.

-58-63  xy

nbsp left parenthesis negative 2 comma negative 2 right parenthesis  (−2,−2)

left parenthesis negative 1 comma negative 4 right parenthesis(−1,−4)

left parenthesis 3 comma 0 right parenthesis(3,0)

left parenthesis 4 comma 2 right parenthesis(4,2)

left parenthesis 6 comma 1 right parenthesis(6,1)

A coordinate system has a horizontal x-axis labeled from negative 5 to 8 and a vertical y-axis labeled from negative 6 to 3. A curve falls from left to right from a solid plotted point at (negative 2, negative 2) to an open plotted point at (negative 1, negative 4). The curve then rises from left to right passing through a solid plotted point at (3, 0) to an open plotted point at (4, 2) where it again falls from left to right to end at a solid plotted point at (6, 1). All points are labeled with coordinates.

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Part 1

Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The absolute maximum is 

enter your response here.

​(Type an integer or a simplified​ fraction.)

B.

There is no absolute maximum for 

yequals=​f(x).

Your answer is correct.

Part 2

Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The absolute minimum is 

enter your response here.

​(Type an integer or a simplified​ fraction.)

B.

There is no absolute minimum for 

yequals=​f(x).

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.

Learn how to analyze a function graph to find absolute maximum and minimum values, and identify local extrema. This problem involves interpreting plotted points on a graph of a function to determine key points and their significance.

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