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.
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left parenthesis 1 comma 7 right parenthesisleft parenthesis 2 comma 4 right parenthesisleft parenthesis 3 comma 5 right parenthesisleft parenthesis 4 comma 2 right parenthesis

A coordinate system has a horizontal x-axis labeled from 0 to 6 in increments of 3 and a vertical y-axis labeled from 0 to 8 in increments of 4. A curve falls from left to right from a solid plotted point at (1, 7) to a solid plotted point at (2, 4). The curve then rises from left to right to a solid plotted point at (3, 5) where it again falls from left to right to end at a solid plotted point at (4, 2). 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 parenthesis

equals

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 nothing

and f left parenthesis nothing right parenthesis equals nothing

.
​(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 identify the absolute maximum, absolute minimum, and any local extrema (local maxima and minima).

### Given Points:
- (1, 7)
- (2, 4)
- (3, 5)
- (4, 2)

### **Step 1: Identify the Absolute Maximum and Minimum**
- **Absolute Maximum**: This is the highest point on the graph.
  - The highest \( y \)-value is 7 at \( x = 1 \).
  - So, the absolute maximum is \( f(1) = 7 \).

- **Absolute Minimum**: This is the lowest point on the graph.
  - The lowest \( y \)-value is 2 at \( x = 4 \).
  - So, the absolute minimum is \( f(4) = 2 \).

### **Step 2: Identify Local Maxima and Minima**
- **Local Maximum**: A point where the function changes from increasing to decreasing.
  - \( x = 1 \) has the highest \( y \)-value of 7 compared to its neighboring points, but since it is at the boundary and the function decreases immediately after, it is an absolute maximum rather than a local maximum.
  - \( x = 3 \) has a \( y \)-value of 5, which is higher than the values at \( x = 2 \) and \( x = 4 \). This is a local maximum.
  
- **Local Minimum**: A point where the function changes from decreasing to increasing.
  - \( x = 2 \) has a \( y \)-value of 4, which is lower than the values at \( x = 1 \) and \( x = 3 \). This is a local minimum.

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

- The **absolute maximum** is at \( f(1) = 7 \).
- The **absolute minimum** is at \( f(4) = 2 \).
- The **local maximum** is at \( f(3) = 5 \).
- The **local minimum** is at \( f(2) = 4 \).

So the correct answer choice for local maxima would be:
- **B. The local maxima of \( y = f(x) \) are \( f(1) = 7 \) and \( f(3) = 5 \)**.

### **Correct Answer: B.**

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

### 5 Related Questions:
1. What is the difference between a local maximum and an absolute maximum?
2. How do you find local minima and maxima using calculus?
3. Can a function have more than one absolute maximum or minimum?
4. How can we determine if a critical point is a maximum, minimum, or neither using the first derivative test?
5. What role do endpoints play in determining absolute maxima and minima?

**Tip:** Always check the boundary points of the domain when looking for absolute maxima or minima, as they can sometimes hold these values.

Learn how to find absolute maximum, absolute minimum, local maxima, and local minima on a function graph. Analyze specific data points and understand the concepts of extrema in calculus.

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