To solve this problem, we need to analyze the function $f(x)$, its first derivative $f'(x)$, and its second derivative $f''(x)$ at each of the labeled points A, B, C, D, and E. Here's the step-by-step analysis:

1. Identifying Points of Interest:

• A: Point where $f(x)$ is at a local minimum.
• B: Point where $f(x)$ crosses the x-axis (root of the function).
• C: Point where $f(x)$ is at a local maximum.
• D: Another point where $f(x)$ crosses the x-axis.
• E: Point on the downward slope after point $D$.

2. Determining Signs:

• $f(x)$:
  • Positive above the x-axis, Negative below the x-axis, Zero on the x-axis.
• $f'(x)$:
  • Positive when the function is increasing, Negative when decreasing, Zero at local maxima or minima.
• $f''(x)$:
  • Positive for concave up (cup-shaped), Negative for concave down (cap-shaped).

3. Analysis of Points:

• Point A:
  • $f(x)$ is Negative (below x-axis).
  • $f'(x)$ is Zero (local minimum).
  • $f''(x)$ is Positive (concave up).
• Point B:
  • $f(x)$ is Zero (crosses x-axis).
  • $f'(x)$ is Positive (increasing slope).
  • $f''(x)$ is Positive (concave up).
• Point C:
  • $f(x)$ is Positive (above x-axis).
  • $f'(x)$ is Zero (local maximum).
  • $f''(x)$ is Negative (concave down).
• Point D:
  • $f(x)$ is Zero (crosses x-axis).
  • $f'(x)$ is Negative (decreasing slope).
  • $f''(x)$ is Negative (concave down).
• Point E:
  • $f(x)$ is Negative (below x-axis).
  • $f'(x)$ is Negative (decreasing slope).
  • $f''(x)$ is Positive (concave up).

4. Final Signs Table:

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

Questions:

1. What determines whether $f'(x)$ is positive or negative at a point?
2. Why is $f'(x)$ zero at local maxima and minima?
3. How can you tell if $f''(x)$ is positive or negative by looking at the graph?
4. What is the significance of the points where $f(x)$ crosses the x-axis?
5. How do concavity and inflection points relate to the sign of $f''(x)$?

Tip:

When analyzing graphs, always pay close attention to the curvature to determine the sign of the second derivative.