It looks like you've uploaded a graph that involves analyzing the function $f(x)$ and its first and second derivatives at different points labeled A, B, C, D, and E. The chart below the graph asks for information regarding the signs (positive, negative, or zero) of $f(x)$, $f'(x)$ (first derivative), and $f''(x)$ (second derivative) at these points.

Let's analyze the graph step by step based on the curves and the critical points:

1. Point A:

• The function $f(x)$ is positive (since the curve is above the x-axis).
• The slope of the tangent (first derivative $f'(x)$) is positive because the function is increasing.
• The second derivative $f''(x)$ is positive, as the curve is concave up at A.

2. Point B:

• The function $f(x)$ is negative (below the x-axis).
• The slope of the tangent $f'(x)$ is zero because B appears to be a minimum point.
• The second derivative $f''(x)$ is positive, since the curve is concave up (indicating a local minimum).

3. Point C:

• The function $f(x)$ is positive (above the x-axis).
• The slope of the tangent $f'(x)$ is zero at point C because it's a local maximum.
• The second derivative $f''(x)$ is negative, as the curve is concave down at C.

4. Point D:

• The function $f(x)$ is negative (below the x-axis).
• The slope $f'(x)$ is negative because the function is decreasing.
• The second derivative $f''(x)$ is negative because the curve is concave down.

5. Point E:

• The function $f(x)$ is negative (below the x-axis).
• The slope $f'(x)$ is zero at E because it appears to be another minimum point.
• The second derivative $f''(x)$ is positive because the curve is concave up at E.

Based on this analysis, you can complete the table as follows:

Let me know if you'd like further details or explanations on any part of this process!

Relative Questions:

1. What are the key characteristics of critical points (minima and maxima)?
2. How do you determine concavity from the second derivative?
3. Why is the first derivative zero at local maxima and minima?
4. What is the relationship between the sign of the second derivative and concavity?
5. How can you interpret the behavior of a function using its first and second derivatives?

Tip:

When analyzing graphs, always check whether the tangent line is horizontal (slope = 0) at points of interest, which often indicates maxima, minima, or points of inflection.