To determine the truth values for each statement, let's analyze each one based on the properties of $f(x)$ inferred from the graph of $f'(x)$:

1. Statement (i): $f(x)$ has a local maximum at $x = -3$

   • A local maximum for $f(x)$ occurs where $f'(x)$ changes from positive to negative. In the graph, around $x = -3$, $f'(x)$ changes from positive to negative. This suggests that $f(x)$ has a local maximum at $x = -3$.
   • Conclusion: True (1)

2. Statement (ii): $f(x)$ has neither a local minimum nor maximum at $x = 3$

   • At $x = 3$, $f'(x) = 0$ (since the graph of $f'(x)$ touches the x-axis at $x = 3$), but it does not change sign; $f'(x)$ remains negative before and after $x = 3$. This indicates a horizontal inflection point, meaning there is neither a maximum nor a minimum for $f(x)$ at $x = 3$.
   • Conclusion: True (1)

3. Statement (iii): $f(x)$ has a local minimum near $x = -2$

   • A local minimum for $f(x)$ would occur where $f'(x)$ changes from negative to positive. However, near $x = -2$, the graph of $f'(x)$ does not change from negative to positive; instead, it remains negative. This implies there is no local minimum near $x = -2$.
   • Conclusion: False (2)

Based on this analysis, the answers for the statements (in the specified order) are:

1, 1, 2

Would you like further details or have any questions? Here are five related questions for further exploration:

1. Why does the sign change in $f'(x)$ indicate a local maximum or minimum in $f(x)$?
2. How can you determine the exact location of inflection points based on $f'(x)$ alone?
3. What does it mean when $f'(x)$ remains constant (doesn’t change sign) around a particular point?
4. How would the answer change if $f(x)$ were discontinuous at any point in the domain?
5. What additional information could we derive if we had the graph of $f''(x)$?

Tip: Remember that if $f'(x)$ is zero but does not change sign, it generally indicates an inflection point rather than a local extremum.