Let's analyze each part based on the graph shown in the image. I'll go over each question step by step.

(a) Find the interval(s) on which $f$ is increasing.

To determine where $f$ is increasing, we look for intervals where the graph moves upward as $x$ increases. Based on the graph:

• $f$ is increasing on the intervals $(0, 1)$, $(3, 5)$, and $(6, 7)$.

Thus, the answer is: $(0, 1) \cup (3, 5) \cup (6, 7)$

(b) Find the interval(s) on which $f$ is decreasing.

Similarly, $f$ is decreasing where the graph moves downward as $x$ increases:

• $f$ is decreasing on the intervals $(1, 3)$ and $(5, 6)$.

The answer is: $(1, 3) \cup (5, 6)$

(c) Find the interval(s) on which $f$ is concave upward.

The function $f$ is concave upward where the graph appears to "curve upward" like a cup, or where its slope increases. Based on the graph:

• $f$ is concave upward on the intervals $(0, 2)$ and $(5, 7)$.

So, the answer is: $(0, 2) \cup (5, 7)$

(d) Find the interval(s) on which $f$ is concave downward.

The function $f$ is concave downward where the graph appears to "curve downward" like an arch, or where its slope decreases. Based on the graph:

• $f$ is concave downward on the interval $(2, 5)$.

The answer is: $(2, 5)$

(e) Find the coordinates of the inflection point(s).

Inflection points occur where the concavity of $f$ changes from upward to downward or vice versa. By examining the graph, we can identify:

• Inflection points at approximately $(2, 2)$ and $(5, 4)$.

The answer is: $(2, 2), (5, 4)$

Would you like further details on any specific part of this solution?

Here are 5 related questions to deepen understanding:

1. How would you find intervals of increase or decrease from a derivative function instead of a graph?
2. What methods can be used to determine concavity if the graph is not provided?
3. Can you describe how inflection points are related to the second derivative?
4. What is the significance of intervals where a function is concave upward or downward?
5. How would the answers change if additional critical points were found within these intervals?

Tip: When analyzing concavity, use the second derivative $f''(x)$ – it helps determine whether a function is concave up (when $f''(x) > 0$) or concave down (when $f''(x) < 0$).