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The question asks to identify the graph of \( f' \) (the derivative of \( f \)) based on the given graph of \( f \), which resembles a bell curve.

### Analyzing the Given Graph \( f(x) \):
The function \( f(x) \) appears to be a smooth curve with the following characteristics:
1. **Increases** at first, reaching a **maximum** at the peak (around \( x = 0 \)).
2. Then **decreases** symmetrically after the peak.

### Features of the Derivative \( f'(x) \):
1. The derivative \( f'(x) \) will be **positive** when \( f(x) \) is increasing (left of the peak).
2. It becomes **zero** at the peak where \( f(x) \) has a maximum.
3. After the peak, as \( f(x) \) decreases, \( f'(x) \) becomes **negative**.
4. The derivative function \( f'(x) \) is likely to be symmetric as well due to the symmetry of \( f(x) \).

### Matching the Derivative Graph:
Based on these characteristics, we can conclude:
- The correct graph for \( f'(x) \) should cross the x-axis at \( x = 0 \) (since the slope of \( f(x) \) is zero there) and have positive values before this and negative values after.

Looking at the options:
- **Option B** matches this description**, with the graph rising, crossing the x-axis at \( x = 0 \), and then falling symmetrically. 

Thus, the correct answer is **B**.

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Would you like further details on this or have any additional questions?

Here are five related questions to consider:
1. What does it mean for a function to be increasing or decreasing in terms of its derivative?
2. How do local maxima or minima of a function relate to the zeros of its derivative?
3. Can a function have a derivative that is always positive? What does this imply about the function?
4. How do you determine points of inflection from the graph of a derivative?
5. What is the relationship between concavity and the second derivative?

**Tip**: The points where the derivative crosses the x-axis correspond to critical points of the original function, such as maxima or minima.

The graph of f is given. Which of the below graphs is a graph of its derivative f'?

This calculus problem involves identifying the derivative of a given function by analyzing its graph. Learn how to interpret the behavior of f(x) and match it with its derivative. Suitable for Grades 11-12 students.

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