Let's analyze the problem.

We are given the graph of $f'(x)$, the derivative of a function $f(x)$, and we are tasked to identify which of the provided options represents the function $f(x)$.

Steps to Solve:

1. Analyze the Graph of $f'(x)$:

   • $f'(x) > 0$ (above the x-axis) for $x > 0$, which implies $f(x)$ is increasing for $x > 0$.
   • $f'(x) < 0$ (below the x-axis) for $x < 0$, which implies $f(x)$ is decreasing for $x < 0$.
   • $f'(x) = 0$ at $x = 0$, indicating a critical point of $f(x)$ (could be a local minimum, maximum, or saddle point).

2. Behavior of $f'(x)$:

   • As $x \to \infty$, $f'(x) \to 0^+$, so $f(x)$ approaches a horizontal asymptote from below.
   • As $x \to -\infty$, $f'(x) \to 0^-$, so $f(x)$ approaches a horizontal asymptote from above.

3. Identify the Shape of $f(x)$:

   • $f(x)$ decreases for $x < 0$, increases for $x > 0$, and has a local minimum at $x = 0$.

4. Match with the Options:

   • Option (a): This shows a curve with a local minimum at $x = 0$ and behavior that matches the analysis above. It correctly reflects $f'(x)$ changing from negative to positive at $x = 0$.
   • Other Options: (b), (c), and (d) do not satisfy the conditions. For example:
     • (b) represents a linear function, which doesn't match.
     • (c) has the wrong curvature and behavior.
     • (d) represents a parabola opening downward, which also doesn't fit.

Correct Answer: (a)

Do you want a more detailed explanation of any specific part of this solution?

Related Questions:

1. What is the relationship between $f(x)$ and $f'(x)$ when $f'(x) > 0$?
2. How do we determine the critical points of $f(x)$ using $f'(x)$?
3. What does it mean if $f'(x) \to 0$ as $x \to \infty$?
4. How does the second derivative $f''(x)$ help in identifying concavity?
5. What types of functions have horizontal asymptotes?

Tip:

For a smooth function $f(x)$, always analyze $f'(x)$ for intervals of increase/decrease and $f''(x)$ for concavity to understand the overall shape of the graph.