To solve this problem, let’s analyze each statement based on the graph of $f'(x)$ provided:

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

   • To have a local maximum at $x = 1$, $f(x)$ should change from increasing to decreasing at that point. This means $f'(x)$ would go from positive to negative as $x$ crosses 1.
   • From the graph of $f'(x)$, we see that $f'(x)$ crosses the x-axis from positive to negative at $x = 1$. This indicates that $f(x)$ has a local maximum at $x = 1$.
   • Statement (i) is True.

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

   • A local minimum or maximum at $x = 3$ would require $f'(x)$ to be zero or undefined there.
   • On the graph, $f'(x)$ touches the x-axis at $x = 3$, meaning $f'(3) = 0$. However, $f'(x)$ does not change sign around $x = 3$; it remains negative on both sides.
   • Since there is no sign change, $x = 3$ is not a point of local extremum for $f(x)$.
   • Statement (ii) is False.

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

   • To confirm a local minimum at $x = -3$, $f'(x)$ should change from negative to positive as $x$ crosses -3.
   • Observing the graph, $f'(x)$ crosses the x-axis from negative to positive at $x = -3$, indicating a local minimum for $f(x)$ at this point.
   • Statement (iii) is True.

Conclusion:

The correct answers are:

• (i) True
• (ii) False
• (iii) True

So, the answer is: 1, 2, 1

Would you like more details on any of these statements?

Follow-Up Questions:

1. What would it mean for $f(x)$ if $f'(x)$ is always positive on an interval?
2. How would you identify an inflection point by looking at the graph of $f'(x)$?
3. Why is it significant that $f'(x)$ touches but does not cross the x-axis at $x = 3$?
4. What additional information could we derive about $f(x)$ from a graph of $f''(x)$?
5. How do the critical points of $f(x)$ relate to the zeros of $f'(x)$?

Tip:

When analyzing the graph of $f'(x)$, remember that a zero crossing of $f'(x)$ with a change in sign indicates a local extremum (maximum or minimum) in $f(x)$.