To solve this problem using the graph of $f(x)$, we need to identify the largest open intervals where the function is either increasing or decreasing.

(a) Increasing Interval:

A function is increasing on an interval when its slope is positive (i.e., the graph is going up as we move from left to right).

• From the graph, $f(x)$ is increasing from $x = 0$ to $x = 3$.
• After $x = 3$, the graph starts decreasing, so the largest open interval where $f(x)$ is increasing is $(0, 3)$.

(b) Decreasing Interval:

A function is decreasing on an interval when its slope is negative (i.e., the graph is going down as we move from left to right).

• The graph of $f(x)$ is decreasing from $x = 3$ to $x = 6$.
• It increases again slightly, but then starts decreasing once more from $x = 7$ to $x = 10$.
• So, the largest open decreasing interval is $(3, 6)$, because it has a larger length than $(7, 10)$.

Final Answers:

• (a) Largest open interval where $f$ is increasing: $(0, 3)$
• (b) Largest open interval where $f$ is decreasing: $(3, 6)$

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of critical points on the graph of $f(x)$?
2. How can you find intervals of increase or decrease without using a graph?
3. What role do local maxima and minima play in determining increasing or decreasing intervals?
4. How would you apply the first derivative test to verify increasing and decreasing intervals?
5. How does concavity relate to intervals of increase or decrease?

Tip: When identifying increasing or decreasing intervals, focus on the direction the graph is moving (up or down) between critical points.