Based on the images you uploaded, let us analyze the graph of $f'(x)$ and verify the given statements about $f(x)$:

Observations:

1. Increasing/Decreasing behavior of $f(x)$:

   • $f(x)$ increases where $f'(x) > 0$ and decreases where $f'(x) < 0$.
   • From the graph of $f'(x)$, we see $f'(x) > 0$ on the intervals: $(-\infty, -6) \quad \text{and} \quad (0, 6).$
   • $f'(x) < 0$ on the intervals: $(-6, 0) \quad \text{and} \quad (6, \infty).$

2. Concavity of $f(x)$:

   • $f(x)$ is concave up where $f''(x) = f'(x)$ is increasing ($f'(x)$ has a positive slope).
   • $f(x)$ is concave down where $f''(x) = f'(x)$ is decreasing ($f'(x)$ has a negative slope).
   • From the graph, $f'(x)$ is:
     • Increasing on $(-2\sqrt{3}, 2\sqrt{3})$, so $f(x)$ is concave up on this interval.
     • Decreasing on $(-\infty, -2\sqrt{3}) \cup (2\sqrt{3}, \infty)$, so $f(x)$ is concave down on these intervals.

3. Critical Points of $f(x)$:

   • Critical points occur where $f'(x) = 0$, which corresponds to $x = -6, 0, 6$.

4. Inflection Points of $f(x)$:

   • Inflection points occur where the concavity changes (i.e., where $f'(x)$ changes from increasing to decreasing or vice versa).
   • This occurs at $x = -2\sqrt{3}$ and $x = 2\sqrt{3}$.

Statement Validation:

1. $f(x)$ is increasing on $(-\infty, -6) \cup (0, 6)$:

   • True. $f'(x) > 0$ on these intervals.

2. $f(x)$ is decreasing on $(-\infty, -6) \cup (0, 6)$:

   • False. $f(x)$ is decreasing on $(-6, 0) \cup (6, \infty)$, not the given intervals.

3. $f(x)$ is concave up on $(-2\sqrt{3}, 2\sqrt{3})$:

   • True. $f'(x)$ is increasing in this interval.

4. $f(x)$ is concave down on $(-2\sqrt{3}, 2\sqrt{3})$:

   • False. $f(x)$ is concave down on $(-\infty, -2\sqrt{3}) \cup (2\sqrt{3}, \infty)$.

5. The critical points of $f(x)$ are $x = -6, x = 0, x = 6$:

   • True. These are the points where $f'(x) = 0$.

6. The inflection points of $f(x)$ are $x = -6, x = 0, x = 6$:

   • False. Inflection points occur at $x = -2\sqrt{3}$ and $x = 2\sqrt{3}$.

Correct Answer:

The correct set of statements is: $\boxed{(1), (3), (5)}$

Would you like further details or clarifications on any step? Here are some additional questions for deeper understanding:

1. Why does $f(x)$ increase when $f'(x) > 0$?
2. How do we determine concavity from the graph of $f'(x)$?
3. What distinguishes a critical point from an inflection point?
4. Why is $f(x)$ concave up specifically on $(-2\sqrt{3}, 2\sqrt{3})$?
5. How can we confirm the exact intervals of increasing and decreasing for $f(x)$ using a test point method?

Tip: Always analyze the behavior of the derivative and its slope to confirm intervals of concavity and