1. **Problem Statement:** Given the graph of a function $f$ from $x=0$ to $x=6$, determine the intervals where $f$ is increasing, decreasing, concave upward, concave downward, and find the inflection points. 2. **Key Concepts:** - A function is **increasing** where its slope (derivative) is positive. - It is **decreasing** where its slope is negative. - The function is **concave upward** where its second derivative is positive (graph curves upward). - It is **concave downward** where its second derivative is negative (graph curves downward). - **Inflection points** occur where the concavity changes, i.e., where the second derivative changes sign. 3. **Analyzing the graph:** - From $x=0$ to $x=1$, $f$ increases sharply (slope positive). - From $x=1$ to $x=3$, $f$ decreases (slope negative). - From $x=3$ to $x=6$, $f$ increases gradually (slope positive). 4. **Intervals of increase and decrease:** - Increasing: $(0,1)$ and $(3,6)$ - Decreasing: $(1,3)$ 5. **Concavity:** - From $x=0$ to about $x=2$, the graph curves upward (concave up). - From about $x=2$ to $x=4$, the graph curves downward (concave down). - From about $x=4$ to $x=6$, the graph curves upward again (concave up). 6. **Intervals of concavity:** - Concave upward: $(0,2)$ and $(4,6)$ - Concave downward: $(2,4)$ 7. **Inflection points:** - At $x=2$, concavity changes from up to down. - At $x=4$, concavity changes from down to up. 8. **Coordinates of inflection points:** - Approximate $f(2)$ and $f(4)$ from the graph: - $f(2) \approx 2$ - $f(4) \approx 3$ **Final answers:** (a) Increasing on $(0,1) \cup (3,6)$ (b) Decreasing on $(1,3)$ (c) Concave upward on $(0,2) \cup (4,6)$ (d) Concave downward on $(2,4)$ (e) Inflection points at $(2,2)$ and $(4,3)$