Math Problem Statement

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. 0 3 6 4 8 x y 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. Question content area bottom 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).

Solution

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 yy-value is 7 at x=1x = 1.
    • So, the absolute maximum is f(1)=7f(1) = 7.
  • Absolute Minimum: This is the lowest point on the graph.

    • The lowest yy-value is 2 at x=4x = 4.
    • So, the absolute minimum is f(4)=2f(4) = 2.

Step 2: Identify Local Maxima and Minima

  • Local Maximum: A point where the function changes from increasing to decreasing.

    • x=1x = 1 has the highest yy-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=3x = 3 has a yy-value of 5, which is higher than the values at x=2x = 2 and x=4x = 4. This is a local maximum.
  • Local Minimum: A point where the function changes from decreasing to increasing.

    • x=2x = 2 has a yy-value of 4, which is lower than the values at x=1x = 1 and x=3x = 3. This is a local minimum.

Step 3: Final Answer Choices

Based on the analysis:

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

So the correct answer choice for local maxima would be:

  • B. The local maxima of y=f(x)y = f(x) are f(1)=7f(1) = 7 and f(3)=5f(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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Extrema (Maxima and Minima)
Graph Analysis

Formulas

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Theorems

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Suitable Grade Level

High School