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A.[-6,4] and [2,5] B.[-6,0] and [2,5] C.(-4,2) D(0,2)

The multiple-choice options provided are intervals where the function \( f(x) \) could be increasing. Let's re-examine the graph in light of these choices.

### Key Observations:
- **Interval \([-6, -4]\)**: The function appears to be increasing from \( x = -6 \) to \( x = -4 \).
- **Interval \((0, 2)\)**: The function is increasing from \( x = 0 \) to \( x = 2 \).
- **Interval \((4, 7)\)**: The function is also increasing from \( x = 4 \) to \( x = 7 \).

### Reviewing the Choices:
- **Option A: \([-6, 4]\) and \([2, 5]\)**  
  The interval \([-6, 4]\) includes both increasing and decreasing parts. The interval \([2, 5]\) includes a decreasing interval. This option is not correct.

- **Option B: \([-6, 0]\) and \([2, 5]\)**  
  The interval \([-6, 0]\) includes a decreasing part. The interval \([2, 5]\) also includes a decreasing interval. This option is not correct.

- **Option C: \((-4, 2)\)**  
  This interval includes both decreasing and increasing parts. It is not entirely increasing, so this option is incorrect.

- **Option D: \((0, 2)\)**  
  The function is indeed increasing between \( x = 0 \) and \( x = 2 \). This is the correct interval.

### Conclusion:
The correct answer is **Option D: \((0, 2)\)**.

Would you like further details or have any questions?

### Follow-up Questions:
1. How do we mathematically confirm that a function is increasing on an interval?
2. What is the role of the first derivative in determining increasing or decreasing intervals?
3. How can the behavior at the endpoints of an interval influence the interval of increase?
4. Can you provide a scenario where a function has both increasing and decreasing intervals within the same range?
5. How does concavity relate to increasing and decreasing functions?

### Tip:
Always double-check the given intervals against the graph to ensure that the function's behavior matches the description in the choices.

Explore how to identify increasing intervals of a function using given intervals and graphical analysis. This problem focuses on analyzing intervals where the function is increasing and selecting the correct interval from multiple choices. Suitable for high school students studying functions and graph analysis.

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