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.

04836xy

left parenthesis 1 comma 3 right parenthesis(1,3)

left parenthesis 3 comma 1 right parenthesis(3,1)

left parenthesis 5 comma 5 right parenthesis(5,5)

left parenthesis 6 comma 3 right parenthesis(6,3)

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

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Part 1

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

A.

The local maximum of 

yequals=​f(x) 

is 

f left parenthesis nothing right parenthesisfenter your response hereequals=enter your response here.

​(Type integers or simplified​ fractions.)

B.

The local maxima of 

yequals=​f(x) 

are 

f left parenthesis nothing right parenthesis equals nothingfenter your response here=enter your response here 

and 

f left parenthesis nothing right parenthesis equals nothingfenter your response here=enter your response here.

​(Type integers or simplified​ fractions.)

C.

There is no local maximum for 

yequals=​f(x).

Let's analyze the given graph and data points to determine the absolute maximum, absolute minimum, and any local extrema (local maxima and minima).

### Given Points:
- (1, 3)
- (3, 1)
- (5, 5)
- (6, 3)

### **Step 1: Identify the Absolute Maximum and Minimum**
- **Absolute Maximum**: This is the highest point on the graph.
  - The highest \( y \)-value is 5 at \( x = 5 \).
  - So, the absolute maximum is \( f(5) = 5 \).

- **Absolute Minimum**: This is the lowest point on the graph.
  - The lowest \( y \)-value is 1 at \( x = 3 \).
  - So, the absolute minimum is \( f(3) = 1 \).

### **Step 2: Identify Local Maxima and Minima**
- **Local Maximum**: A point where the function changes from increasing to decreasing.
  - \( x = 5 \) has a \( y \)-value of 5, which is higher than the values at \( x = 3 \) and \( x = 6 \). This is a local maximum.
  
- **Local Minimum**: A point where the function changes from decreasing to increasing.
  - \( x = 3 \) has a \( y \)-value of 1, which is lower than the values at \( x = 1 \) and \( x = 5 \). This is a local minimum.

### **Step 3: Final Answer Choices**
Based on the analysis:

- The **absolute maximum** is at \( f(5) = 5 \).
- The **absolute minimum** is at \( f(3) = 1 \).
- The **local maximum** is at \( f(5) = 5 \).
- The **local minimum** is at \( f(3) = 1 \).

So the correct answer choice for local maxima would be:
- **A. The local maximum of \( y = f(x) \) is \( f(5) = 5 \).**

### **Correct Answer: A.**

**Do you need any further details or have any questions?**

### 5 Related Questions:
1. What is the distinction between absolute and local extrema in a function?
2. How do you determine the local maximum or minimum points using the second derivative test?
3. Can a function have both local and absolute maximum at the same point?
4. What role does symmetry play in identifying maximum and minimum points on a graph?
5. How do changes in the function's domain affect the location of absolute maxima and minima?

**Tip:** When analyzing a graph, pay close attention to the direction of the curve at critical points, as this will help identify whether they are maxima, minima, or neither.

Explore how to find absolute maximum, absolute minimum, and local extrema of a function using given data points and graph analysis. Includes step-by-step analysis and answer choices for identifying local maxima and minima.

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