To address this question, we need to analyze the provided graph of $f(x)$ to determine where its first derivative $f'(x)$ and second derivative $f''(x)$ are positive or negative.

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

• $f'(x) > 0$ where the graph of $f(x)$ is increasing. From the graph:
  • The function increases on $(0, 2)$ and $(4, 6)$.

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

• $f'(x) < 0$ where the graph of $f(x)$ is decreasing. From the graph:
  • The function decreases on $(-\infty, 0)$, $(2, 4)$, and $(6, \infty)$.

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

• $f''(x) > 0$ where the graph of $f(x)$ is concave up. From the graph:
  • The function is concave up on $(3, 5)$.

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

• $f''(x) < 0$ where the graph of $f(x)$ is concave down. From the graph:
  • The function is concave down on $(1, 3)$ and $(5, 7)$.

Would you like a detailed explanation of how to interpret the derivatives using the graph?

5 Related Questions:

1. How do you determine intervals of increase and decrease from a graph?
2. What is the role of the first derivative in finding the slope of a function?
3. How can concavity be visually identified from a graph?
4. Why is the second derivative related to concavity changes?
5. How are critical points related to the first derivative?

Tip: To ensure correct interval notation, always use open intervals for behavior that excludes endpoints.