1. **Problem Statement:** Given the graph of the derivative function $f'(x)$, classify two different key features on $f'(x)$ and determine the corresponding key features on the original function $f(x)$. 2. **Key Concept:** The relationship between $f'(x)$ and $f(x)$ is as follows: - Where $f'(x) = 0$ and changes sign, $f(x)$ has a local maximum or minimum (critical points). - Where $f'(x)$ has a local maximum or minimum, $f(x)$ has an inflection point. 3. **Feature 1:** At $x=0$, $f'(x)$ touches the x-axis and has a local maximum near $(0,0)$. - Since $f'(0) = 0$ and $f'(x)$ changes from positive to negative, $f(x)$ has a local maximum at $x=0$. 4. **Feature 2:** At $x=2$, $f'(x)$ has a local minimum near $(2,-16)$. - Since $f'(x)$ has a local minimum here, $f(x)$ has an inflection point at $x=2$. 5. **Summary Table:** | $x$-value | Key feature on $f'(x)$ | Corresponding key feature on $f(x)$ | |---|---|---| | 0 | $f'(x)$ zero crossing with local max | Local maximum | | 2 | Local minimum of $f'(x)$ | Inflection point | Thus, the two key features on $f'(x)$ are the zero crossing at $x=0$ (local max) and the local minimum at $x=2$, corresponding to a local maximum and an inflection point on $f(x)$ respectively.