To solve the problem, let's analyze the given graph of $f(x)$ and determine the intervals for the derivatives $f'(x)$ and $f''(x)$.

Step-by-Step Solution:

Part a) On which interval(s) is $f'(x) > 0$?

The derivative $f'(x) > 0$ when the graph of $f(x)$ is increasing. Observing the graph:

• $f(x)$ increases from $x = 1$ to $x = 3$, and from $x = 5$ to $x = 7$.

Thus, $f'(x) > 0$ on: $(1, 3), (5, 7)$

Part b) On which interval(s) is $f'(x) < 0$?

The derivative $f'(x) < 0$ when the graph of $f(x)$ is decreasing. Observing the graph:

• $f(x)$ decreases from $x = 3$ to $x = 5$.

Thus, $f'(x) < 0$ on: $(3, 5)$

Part c) On which interval(s) is $f''(x) > 0$?

The second derivative $f''(x) > 0$ when the graph of $f(x)$ is concave up. Observing the graph:

• $f(x)$ is concave up from $x = 4$ to $x = 7$.

Thus, $f''(x) > 0$ on: $(4, 7)$

Part d) On which interval(s) is $f''(x) < 0$?

The second derivative $f''(x) < 0$ when the graph of $f(x)$ is concave down. Observing the graph:

• $f(x)$ is concave down from $x = 1$ to $x = 4$.

Thus, $f''(x) < 0$ on: $(1, 4)$

Final Answers:

1. $f'(x) > 0$: $(1, 3), (5, 7)$
2. $f'(x) < 0$: $(3, 5)$
3. $f''(x) > 0$: $(4, 7)$
4. $f''(x) < 0$: $(1, 4)$

Do you want further clarification or a detailed breakdown of any interval?

Relative Questions to Expand Understanding:

1. How can the concepts of increasing and decreasing functions be related to $f'(x) > 0$ or $f'(x) < 0$?
2. What is the significance of concavity in a graph, and how does it connect to $f''(x)$?
3. How would the answers change if the graph had sharper changes in slope or additional turning points?
4. How do we find intervals for $f'(x) = 0$, and what does that signify on a graph?
5. What are practical applications of determining the intervals where $f'(x) > 0$ or $f''(x) > 0$?

Tip:

Always pay attention to the turning points of $f(x)$ (where $f'(x) = 0$) and the inflection points (where $f''(x) = 0$) when analyzing graphs.