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.