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.