https://file.aiquickdraw.com/mathbot/file/1727499237418ozvxnt7z.jpg

To solve this problem, let's analyze the graph of \( f(x) \) provided and the corresponding graphs for its derivative \( f'(x) \).

### Key Observations About \( f(x) \):
1. **Shape of \( f(x) \)**: The graph of \( f(x) \) has a smooth, bell-curve shape. It rises to a peak and then symmetrically decreases. 
2. **Critical Points**: 
   - At the peak (maximum), the slope is zero, so \( f'(x) = 0 \) at that point.
   - As \( x \) approaches the peak from the left, \( f(x) \) is increasing, so \( f'(x) \) should be positive in this region.
   - As \( x \) moves to the right of the peak, \( f(x) \) is decreasing, so \( f'(x) \) should be negative here.

### Characteristics of \( f'(x) \):
- \( f'(x) \) must be zero at the peak of the bell curve.
- \( f'(x) \) must be positive before the peak and negative after the peak.
- The transition from positive to negative suggests a point where the graph of \( f'(x) \) crosses the x-axis.

### Analysis of the Graphs:
Looking at the available options (A) through (H):
- The correct graph for \( f'(x) \) should cross the x-axis, have positive values to the left of the peak and negative values to the right.

From this, the graph in **Option (B)** is the most suitable because:
- It shows a zero at the point where \( f(x) \) has its maximum.
- The graph is positive on the left and negative on the right of the zero point, matching the behavior of \( f(x) \).

Thus, **Option (B)** is the graph of the derivative \( f'(x) \).

Would you like more details or have any other questions?

### Related Questions:
1. How do we determine if a graph represents the derivative of a function?
2. What does the slope of a function at a point tell us about its derivative?
3. How can we recognize inflection points on a graph of a derivative?
4. How would the second derivative \( f''(x) \) look for this function?
5. Why is the derivative zero at the peak of a function?

### Tip:
When analyzing derivatives, always pay attention to where the original function is increasing, decreasing, or flat (critical points), as this directly influences the behavior of the derivative graph.

The graph of f(x) is given. Which of the below graphs is the graph of its derivative f'(x)?

Solve this calculus problem by analyzing the graph of f(x) and determining the correct derivative graph based on critical points and the behavior of the function. This problem covers key calculus concepts, including derivatives and the First Derivative Test, suitable for Grades 11-12.

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