To solve this problem, let’s analyze the provided graph and statements based on the properties of $f(x)$ and its derivative $f'(x)$.

Analysis of Given Information

• We are provided with a graph of $f'(x)$, the derivative of $f(x)$.
• Since $f(x)$ is continuous, we know that $f'(x)$ indicates the slope of $f(x)$ at each point.
• The sign changes in $f'(x)$ help identify local maxima and minima in $f(x)$:
  • If $f'(x)$ changes from positive to negative at a point, $f(x)$ has a local maximum at that point.
  • If $f'(x)$ changes from negative to positive at a point, $f(x)$ has a local minimum at that point.

Analysis of Each Statement

Let's evaluate each statement individually.

1. Statement (i): $f(x)$ has a local maximum at $x = 1$

   • At $x = 1$, $f'(x)$ crosses the x-axis from positive to negative.
   • This indicates that $f(x)$ has a local maximum at $x = 1$.
   • Conclusion: True.

2. Statement (ii): $f(x)$ must have a local minimum or maximum at $x = 3$

   • At $x = 3$, $f'(x)$ touches the x-axis but does not change sign; it remains negative before and after $x = 3$.
   • This indicates that $x = 3$ is a critical point (where $f'(x) = 0$), but it is not a point of local maximum or minimum because the slope does not change from positive to negative or vice versa.
   • Conclusion: False.

3. Statement (iii): $f(x)$ has a local minimum at $x = -3$

   • At $x = -3$, $f'(x)$ crosses the x-axis from negative to positive.
   • This indicates that $f(x)$ has a local minimum at $x = -3$.
   • Conclusion: True.

Final Answer

Based on the analysis, the answers to the statements are:

1. True
2. False
3. True

Thus, the correct answer is: 1, 2, 1

Would you like further details on any part of the solution or have any questions?

Related Questions:

1. What is the significance of the derivative sign change in determining local extrema?
2. How can we distinguish between a critical point and an extremum using the first derivative?
3. What other methods can be used to confirm the nature of critical points besides the first derivative test?
4. How does the continuity of $f(x)$ affect the behavior of $f'(x)$ at critical points?
5. Can points where $f'(x) = 0$ but does not change sign have any particular interpretations in real-world applications?

Tip:

When analyzing graphs of $f'(x)$, remember that sign changes in $f'(x)$ indicate local maxima and minima in $f(x)$, as this directly reflects the slope behavior of the original function.