Final answer:
To determine the characteristics of labeled points on a graph, analyze critical points, apply derivative tests, discuss extrema characteristics, and evaluate concavity.
Step-by-step explanation:
A. Analyze the critical points on the graph:
Critical points occur where the derivative of the function is zero or undefined. We can find these points by setting the derivative equal to zero and solving for x.
B. Apply the first and second derivative tests for extrema:
Use the first derivative test by evaluating the sign of the derivative at points of interest to determine the relative extrema. Then, use the second derivative test to determine the concavity of the function at those points and identify possible points of inflection.
C. Discuss the characteristics of absolute and relative extrema:
Absolute extrema are the highest or lowest points on the entire graph, while relative extrema are the highest or lowest points within a specific interval.
D. Evaluate the concavity of the function at labeled points:
The concavity of the function can be determined by analyzing the second derivative. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.