1. **State the problem:** We need to determine where the function $f$ is increasing and decreasing, identify the concavity intervals (where it is concave up or down), and find the points of inflection. 2. **Interpret the graph description:** - The function starts high when $x$ is near -1 and decreases until close to $x=1$. - There is a local minimum near $x=1$. - The function then increases to a local maximum near $x=3.5$. - After that, it decreases to a local minimum near $x=5.5$. - Finally, it increases steadily for $x > 5.5$. 3. **Intervals where $f$ is increasing or decreasing:** - Increasing means the function goes upward as $x$ increases. - Decreasing means the function goes downward as $x$ increases. From the graph description: - Increasing on $(1, 3.5)$ and $(5.5, \infty)$ - Decreasing on $(-1, 1)$ and $(3.5, 5.5)$ 4. **Concavity and inflection points:** - Concave up where the graph curves upward (like a cup). - Concave down where the graph curves downward (like a cap). - Points of inflection occur where the concavity changes from up to down or down to up. From the curve: - It is concave down before $x=1$, then concave up between about $x=1$ and $x=3.5$. - Concave down between $x=3.5$ and $x=5.5$. - Concave up after $x=5.5$. Therefore, points of inflection are approximately at $x=1$, $x=3.5$, and $x=5.5$. **Final answers:** - Increasing intervals: $(1, 3.5) \cup (5.5, \infty)$ - Decreasing intervals: $(-1, 1) \cup (3.5, 5.5)$ - Points of inflection: $1, 3.5, 5.5$