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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).

The graph of f', the derivative of the function f, is shown above for -3 ≤ x ≤ 3. On what intervals is f increasing?

This problem involves determining where the function f is increasing based on the graph of its derivative f'. The key mathematical concepts include calculus, derivatives, and intervals of increase and decrease. Learn how to analyze the graph and apply the First Derivative Test to find the correct intervals. Suitable for Grades 11-12.

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