To determine where the function $f$ is concave upward or concave downward, we examine the curvature of the graph as it progresses. Concavity of a function changes at inflection points, where the graph changes from concave up to concave down, or vice versa. A graph is concave upward when it bends upward (like a cup) and concave downward when it bends downward (like a hill).

From the description, the key inflection points seem to occur around $x = 1$ and $x = 3$, where the graph sharply changes direction. Here’s a breakdown of the behavior:

1. Concave Upward: The graph is concave upward on intervals where it appears to bend upwards between inflection points. This would likely be in intervals: $(1, 2) \cup (3, \infty)$

2. Concave Downward: The graph is concave downward on intervals where it bends downward between inflection points. This would likely be in intervals: $(-\infty, 1) \cup (2, 3)$

Thus, the solution is:

• Concave upward: $(1, 2) \cup (3, \infty)$
• Concave downward: $(-\infty, 1) \cup (2, 3)$

Let me know if you need further details or have any questions!

Here are some related questions to consider:

1. How do inflection points relate to the concavity of a function?
2. Can concavity intervals be determined without a graph?
3. What is the relationship between concavity and the second derivative?
4. How does one identify maximum and minimum points using the first and second derivatives?
5. How would concavity affect the optimization of real-world functions?

Tip: Inflection points, where concavity changes, often correspond to where the second derivative equals zero or is undefined.