1. **Problem statement:** We are given the graph of $f'$, the derivative of $f$, for $x>0$. We need to determine which of the statements I, II, and III are true: I. $f(2) < f(5) < f(8)$ II. $f'(2) < f'(5) < f'(8)$ III. $f''(2) < f''(5) < f''(8)$ 2. **Understanding the graph:** The graph of $f'$ starts high near $y=25$ at $x=0$ and decreases rapidly, approaching the $x$-axis asymptotically as $x$ increases to 10. This means: - $f'(x)$ is positive but decreasing for $x>0$. - Since $f'(x)$ is positive, $f(x)$ is increasing. - Since $f'(x)$ is decreasing, $f''(x) = (f')'(x)$ is negative. 3. **Analyze statement I: $f(2) < f(5) < f(8)$** - Because $f'(x) > 0$ for all $x>0$, $f$ is increasing. - Therefore, $f(2) < f(5) < f(8)$ is true. 4. **Analyze statement II: $f'(2) < f'(5) < f'(8)$** - The graph of $f'$ is decreasing, so $f'(2) > f'(5) > f'(8)$. - Hence, $f'(2) < f'(5) < f'(8)$ is false. 5. **Analyze statement III: $f''(2) < f''(5) < f''(8)$** - Since $f'(x)$ is decreasing and concave up (approaching zero asymptotically), $f''(x) = (f')'(x)$ is negative. - The slope of $f'$ is increasing (becoming less negative) as $x$ increases, so $f''(x)$ is increasing. - Therefore, $f''(2) < f''(5) < f''(8)$ is true. 6. **Conclusion:** Statements I and III are true, II is false. **Final answer:** C) I and III only