Question

How to check for absolute maximum and minimum:

a) Set the derivative equal to zero
b) Evaluate at critical points
c) Use the first derivative test
d) Apply the second derivative test

Answer 1

To check for absolute maxima and minima, find and evaluate critical points, use the first and second derivative tests, and evaluate the function at the interval's endpoints.

Step-by-step explanation:

To check for absolute maximum and minimum values of a function on a given interval, you can follow these steps:

1. Set the derivative of the function equal to zero to find potential critical points.
2. Evaluate the function at these critical points to determine their values.
3. Use the first derivative test to analyze the sign changes of the derivative around these points, which helps identify whether each point is a maximum or minimum.
4. Apply the second derivative test, by taking the second derivative of the function and evaluating it at the critical points. If the second derivative is positive at a critical point, this indicates a local minimum; if it's negative, it indicates a local maximum; and if it's zero, the test is inconclusive.

It's important to also evaluate the function at the endpoints of the interval, as absolute maxima or minima can occur there as well.