Looking at the graph provided in your image, let's break down how to solve the problem step by step.

Understanding the Graph:

• You have a function $f(x)$ graphed, with clear points marked on the graph:
  • A local maximum at $(3, 4)$
  • A local maximum at $(2, 5)$
  • A local minimum at $(-2, 1)$
  • Another minimum at $(0, 0)$

Question Breakdown:

1. Local Maximum:

   • The graph shows two peaks where the function attains a local maximum.
   • The local maxima occur at $x = 2$ and $x = 3$, with corresponding $f(x)$ values of $5$ and $4$, respectively.
   • Answer for part B (Local Maxima): $f(2) = 5$ and $f(3) = 4$.

2. Local Minimum:

   • The graph has two troughs (local minima).
   • The local minima occur at $x = -2$ and $x = 0$, with corresponding $f(x)$ values of $1$ and $0$.
   • Answer for part B (Local Minima): $f(-2) = 1$ and $f(0) = 0$.

Correct Selection:

• For Local Minima, the correct answer would be B, where $f(0) = 0$ and $f(-2) = 1$.

Full Answer Recap:

• Local maxima: $f(2) = 5$, $f(3) = 4$.
• Local minima: $f(-2) = 1$, $f(0) = 0$.

Would you like further details on any part of this process?

Here are 5 related questions to help deepen your understanding:

1. What is the difference between local and absolute maxima and minima?
2. How can we determine if a critical point is a local max or min using the derivative?
3. What does the second derivative tell us about concavity at these points?
4. How would the behavior change if there were more turning points?
5. Can a function have no local maxima or minima, and if so, under what conditions?

Tip: Always look for the behavior of the function around critical points to determine local extrema.