Final answer:
To find the absolute maximum and minimum of a function on a given interval, apply critical point analysis, utilize the endpoints of the interval, and use the first and second derivative tests, then compare all identified values.
Step-by-step explanation:
To determine the absolute maximum and minimum of a function on a given interval, you can follow these steps:
- Apply critical point analysis: First, find the function's critical points by setting its first derivative equal to zero and solving for the variable. These are the points where the function may have a potential max or min.
- Utilizing the endpoints of the interval: Evaluate the function's value at the endpoints of the interval, since the absolute maximum or minimum could occur at these points.
- Using the first derivative test: Use the first derivative to test the behavior of the function around each critical point to determine if each is a maximum, minimum, or neither.
- Employing the second derivative test: Alternatively, you can use the second derivative of the function at critical points to determine concavity and hence infer if a critical point is a maximum or minimum.
- Comparing values: Finally, compare the function values at critical points and endpoints to identify the absolute maximum and minimum on the interval.
Remember that the absolute maximum and minimum refer to the highest and lowest values the function reaches on the specified interval, respectively.