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The task asks to match the graph with appropriate intervals based on the average rate of change (keskimääräinen muutosnopeus) of the function in different intervals. The graph shows a curve representing the function \( f(x) \), and the intervals to choose from are:

- \([1, 2]\)
- \([-1, 1]\)
- \([3, 4]\)
- \([2, 3]\)

Let’s analyze the options:

1. **Keskimääräinen muutosnopeus on negatiivinen (The average rate of change is negative):**
   - This means the function is decreasing over the given interval. The graph decreases between \( x = 1 \) and \( x = 2 \), so the correct interval is **[1, 2]**.

2. **Keskimääräinen muutosnopeus on annetuista väleistä suurin tällä välillä (The average rate of change is the greatest in this interval):**
   - The average rate of change is positive and largest when the slope is steepest. Between \( x = 3 \) and \( x = 4 \), the curve is sharply increasing, so the correct interval is **[3, 4]**.

3. **Keskimääräinen muutosnopeus on nolla (The average rate of change is zero):**
   - This occurs when the function has no change, i.e., at a horizontal tangent. From the graph, there is a point where the curve flattens between \( x = -1 \) and \( x = 1 \), so the correct interval is **[-1, 1]**.

4. **Keskimääräinen muutosnopeus on sama kuin välillä [1, 4] (The average rate of change is the same as in the interval [1, 4]):**
   - This asks for an interval where the average rate of change matches the one in \([1, 4]\). In \([2, 3]\), the curve has a similar gentle slope as it does in \([1, 4]\), so the correct interval is **[2, 3]**.

To summarize the answers:
1. [1, 2]: **Keskimääräinen muutosnopeus on negatiivinen.**
2. [3, 4]: **Keskimääräinen muutosnopeus on annetuista väleistä suurin.**
3. [-1, 1]: **Keskimääräinen muutosnopeus on nolla.**
4. [2, 3]: **Keskimääräinen muutosnopeus on sama kuin välillä [1, 4].**

Would you like more details or have any questions? 

Here are 5 related questions:
1. What is the mathematical definition of the average rate of change of a function?
2. How do we calculate the slope of a curve over a given interval?
3. What are the properties of increasing and decreasing functions?
4. How does the concept of a derivative relate to the average rate of change?
5. How can concavity affect the rate of change of a function?

**Tip**: The average rate of change between two points on a graph is equivalent to the slope of the line connecting those points (i.e., the secant line).

Match the given graph with the appropriate statements regarding the average rate of change in the intervals [1, 2], [-1, 1], [3, 4], and [2, 3].

Learn how to analyze the average rate of change of a function from its graph. This problem covers key concepts of increasing and decreasing functions, horizontal tangents, and interpreting graphs.

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