1. **Problem Statement:** We are given three graphs and need to determine which corresponds to the function $f$, its first derivative $f'$, and its second derivative $f''$. 2. **Key Concepts:** - The original function $f$ shows the shape of the curve. - The first derivative $f'$ represents the slope of $f$; where $f$ has local maxima or minima, $f'$ crosses zero. - The second derivative $f''$ indicates concavity; where $f''$ is positive, $f$ is concave up, and where $f''$ is negative, $f$ is concave down. 3. **Analyzing the Top-left graph:** - It looks like a cubic function with two turning points (local min near $x=-6$ and local max near $x=6$). - The function passes through roughly $(0,0)$. - This matches the behavior of the original function $f$. 4. **Analyzing the Top-right graph:** - Shows multiple oscillations with zero crossings near the turning points of $f$. - Since $f'$ crosses zero at the local extrema of $f$, this graph corresponds to $f'$. 5. **Analyzing the Bottom-left graph:** - Has one local max and one local min, indicating changes in concavity. - This matches the second derivative $f''$, which shows where $f$ changes concavity (inflection points). 6. **Summary:** - Top-left graph: $f$ - Top-right graph: $f'$ - Bottom-left graph: $f''$ **Final answer:** - The top-left graph corresponds to $f$. - The top-right graph corresponds to $f'$. - The bottom-left graph corresponds to $f''$.