1. **Problem Statement:** We are given the graph of the first derivative of a function $f$, denoted $f'(x)$. We need to determine which of the given graphs may represent the original function $f(x)$. 2. **Understanding the given $f'(x)$ graph:** The graph of $f'(x)$ has a vertical asymptote at $x=0$ where it tends to positive infinity, and as $x \to +\infty$, $f'(x) \to 0$. This suggests: - Near $x=0$, the slope of $f(x)$ is very large positive. - As $x$ increases, the slope $f'(x)$ decreases towards zero. 3. **Interpreting $f(x)$ from $f'(x)$ behavior:** Since $f'(x)$ is positive from $0$ to $+\infty$ and decreasing toward zero, the original function $f(x)$ is increasing and concave down on this interval. Near $x=0$, because $f'(x)$ tends to $+\infty$, the function $f$ grows very steeply from the left side of zero. 4. **Checking the candidate graphs:** - Graph (a) (bottom-left) starts from the third quadrant (negative $x$, negative $y$), passes through the origin and increases slowly in the first quadrant. This matches a function increasing on $(0, \infty)$ but starting negative and crossing zero. This could correspond if $f$ is defined on all $\mathbb{R}$ and increasing after $0$. - Graph (b) (bottom-center) starts in the second quadrant, crossing through the origin, and increases linearly in the first quadrant, which also fits an increasing function with slope tending to zero in the right half-plane. 5. **Conclusion:** Both graphs (a) and (b) show an increasing function after zero, consistent with $f'(x)$ positive and decreasing to zero on $(0, \infty)$. Therefore, either or both could represent $f(x)$. **Final answer:** The function $f$ may be represented by graph (a) or graph (b) because both comply with the properties indicated by $f'(x)$.