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.

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.