To determine on which intervals the function $f$ is increasing, we need to analyze the graph of $f'$, the derivative of $f$. The function $f$ is increasing where $f'(x) > 0$ (where the derivative is positive).

Let's break this down step by step:

1. The graph of $f'$ is positive (above the x-axis) in the intervals:

   • From $x = -3$ to $x = -1$
   • From $x = 1$ to $x = 3$

2. The graph of $f'$ is negative (below the x-axis) in the intervals:

   • From $x = -1$ to $x = 1$

Thus, $f$ is increasing on the intervals where $f' > 0$, which are:

• $[-3, -1]$ and $[1, 3]$.

Therefore, the correct answer is (D) $[-3, -1]$ and $[1, 3]$.

Would you like further clarification or details?

Here are five questions to expand the understanding of this concept:

1. What happens to the graph of $f$ when $f'(x) = 0$?
2. Can a function have multiple intervals where it is increasing and decreasing? Why?
3. How would the graph of $f$ look based on the behavior of $f'$?
4. What does it mean if $f'(x)$ is constant over an interval?
5. How does the second derivative $f''$ affect the shape of $f$?

Tip: Always check where the derivative changes sign (crosses the x-axis) to identify points where the function changes from increasing to decreasing (or vice versa).