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.