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The problem asks you to identify the graph of the derivative \( f'(x) \) based on the given graph of \( f(x) \), which is shown in blue in the image.

To approach this problem, we need to analyze the shape and behavior of the original graph, \( f(x) \):

- \( f(x) \) is a smooth, bell-shaped curve. It rises steeply to a maximum at \( x = 0 \), then decreases symmetrically.
- The function appears to have:
  - A maximum at \( x = 0 \), meaning \( f'(0) = 0 \).
  - Positive slope for \( x < 0 \) (increasing function), so \( f'(x) > 0 \) on the left of the peak.
  - Negative slope for \( x > 0 \) (decreasing function), so \( f'(x) < 0 \) on the right of the peak.

### Derivative behavior:
- \( f'(x) \) should be zero at \( x = 0 \) (since there’s a local maximum).
- \( f'(x) \) is positive for \( x < 0 \) and becomes negative for \( x > 0 \), meaning that \( f'(x) \) crosses the x-axis at \( x = 0 \).

Looking at the options:
- Graphs (B) and (H) show the behavior of the derivative where \( f'(x) \) is positive for \( x < 0 \), zero at \( x = 0 \), and negative for \( x > 0 \).
- The graph in option (H) resembles the derivative of the bell-shaped curve more closely.

Thus, the correct answer should be **(H)**.

Do you want more details or have any questions?

Here are 5 related questions:
1. How do we interpret the maximum and minimum points of a graph in terms of derivatives?
2. What is the relationship between the concavity of a function and its second derivative?
3. How does the sign of the first derivative \( f'(x) \) inform us about the increasing or decreasing behavior of \( f(x) \)?
4. What are critical points, and how do we identify them using the first derivative?
5. How can we determine if a critical point is a local maximum, minimum, or saddle point?

**Tip**: When analyzing graphs, always pay attention to turning points and inflection points, as these give vital information about the derivative and second derivative.

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

Learn how to determine the graph of the derivative f'(x) from the given graph of the function f(x). The problem covers key calculus concepts such as identifying critical points, analyzing increasing and decreasing intervals, and applying the first derivative test.

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