The image displays two graphs, one solid and one dashed, representing a function $f(x)$ and its derivative $f'(x)$. The question asks us to identify which graph corresponds to $f(x)$ and which to $f'(x)$.

To solve this:

1. Key point: The derivative $f'(x)$ gives us information about the slope of $f(x)$. So, where the graph of $f(x)$ has horizontal tangents (slope = 0), the graph of $f'(x)$ should cross the x-axis (since $f'(x) = 0$).

2. Looking at the solid curve:

   • The solid curve has horizontal tangents (slope = 0) at approximately $x = -4$, $x = 0$, and $x = 4$.
   • Correspondingly, the dashed curve crosses the x-axis at those same points, meaning the derivative $f'(x) = 0$ there.

3. Conclusion:

   • The solid curve represents $f(x)$, and the dashed curve represents its derivative $f'(x)$.

Thus, Option A is correct: $f(x)$ is the solid line, and $f'(x)$ is the dashed line.

Do you need any further explanation or details?

Related Questions:

1. What does it mean when the derivative $f'(x)$ is negative?
2. How do inflection points on $f(x)$ relate to the graph of $f'(x)$?
3. How can you identify concavity of $f(x)$ from the graph of $f'(x)$?
4. What happens to the graph of $f'(x)$ at points where $f(x)$ has a maximum or minimum?
5. How would the graph of the second derivative $f''(x)$ relate to the graphs shown?

Tip:

When analyzing graphs of a function and its derivative, focus on points where the function's slope is zero (horizontal tangents) and where the derivative changes sign (positive to negative or vice versa).