1. **Problem Statement:** Given the graph of the derivative $f'(x)$ of a continuous function $f$, determine intervals where $f$ is increasing or decreasing, find local maxima and minima of $f$, identify intervals of concavity, and locate points of inflection. 2. **Key Concepts:** - $f$ is increasing where $f'(x) > 0$. - $f$ is decreasing where $f'(x) < 0$. - Local maxima occur where $f'(x)$ changes from positive to negative. - Local minima occur where $f'(x)$ changes from negative to positive. - Concavity of $f$ is determined by the sign of $f''(x)$, the derivative of $f'(x)$. - $f$ is concave upward where $f''(x) > 0$ (i.e., $f'(x)$ is increasing). - $f$ is concave downward where $f''(x) < 0$ (i.e., $f'(x)$ is decreasing). - Points of inflection occur where $f''(x)$ changes sign, i.e., where $f'(x)$ changes from increasing to decreasing or vice versa. 3. **Analyzing the graph of $f'(x)$:** - $f'(x)$ crosses the x-axis approximately at $x=1$, $x=3.5$, $x=6.5$, and $x=8.5$. - $f'(x) > 0$ on intervals $(1,3.5)$ and $(8.5, \infty)$ approximately. - $f'(x) < 0$ on intervals $(0,1)$, $(3.5,6.5)$, and $(6.5,8.5)$ approximately. - $f'(x)$ has local maxima near $x=2$ and $x=5$ and a local minimum near $x=7$. 4. **Intervals where $f$ is increasing:** - Where $f'(x) > 0$: approximately $(1,3.5)$ and $(8.5, \infty)$. 5. **Intervals where $f$ is decreasing:** - Where $f'(x) < 0$: approximately $(0,1)$, $(3.5,6.5)$, and $(6.5,8.5)$. 6. **Local maxima of $f$:** - Occur where $f'(x)$ changes from positive to negative. - This happens near $x=3.5$ and $x=8.5$. 7. **Local minima of $f$:** - Occur where $f'(x)$ changes from negative to positive. - This happens near $x=1$ and $x=6.5$. 8. **Concavity of $f$:** - $f$ is concave upward where $f'(x)$ is increasing. - From the graph, $f'(x)$ increases on intervals approximately $(0,2)$ and $(6,8)$. - $f$ is concave downward where $f'(x)$ is decreasing. - From the graph, $f'(x)$ decreases on intervals approximately $(2,6)$ and $(8, \infty)$. 9. **Points of inflection:** - Occur where $f'(x)$ changes from increasing to decreasing or vice versa. - From the graph, these points are near $x=2$, $x=6$, and $x=8$. **Final answers:** - $f$ increasing on intervals: $(1,3.5) \cup (8.5, \infty)$ - $f$ decreasing on intervals: $(0,1) \cup (3.5,6.5) \cup (6.5,8.5)$ - Local maxima at $x = 3.5, 8.5$ - Local minima at $x = 1, 6.5$ - $f$ concave upward on intervals: $(0,2) \cup (6,8)$ - $f$ concave downward on intervals: $(2,6) \cup (8, \infty)$ - Points of inflection at $x = 2, 6, 8$