Question

How do you determine the absolute maximum and minimum values of a function on the given interval?

a) By taking the first derivative and setting it equal to zero
b) By finding critical points and evaluating the function at these points
c) By using the second derivative test
d) By analyzing the function at the endpoints of the interval

Answer 1

To find the absolute maximum and minimum values of a function on a given interval, calculate the derivative, set it to zero to find critical points, evaluate those points and the endpoints of the interval, then compare these values to determine the extrema.

Step-by-step explanation:

To determine the absolute maximum and minimum values of a function on a given interval, you should follow these steps:

1. Find the derivative of the function and set it equal to zero to locate critical points.
2. Evaluate the function at these critical points to find local maxima and minima.
3. Analyze the function at the endpoints of the interval, as absolute extrema could occur here as well.
4. Determine which of the function values from the critical points and endpoints are the largest and smallest. These will be your absolute maximum and minimum on the interval.

Option (b) finding critical points and evaluating the function at these points, and option (d) analyzing the function at the endpoints of the interval are the correct methods to find the absolute maxima and minima.

Option (c), using the second derivative test, can help determine the concavity of the function and the nature (maxima or minima) of the critical points but is not solely sufficient to determine the absolute extrema. However, be aware that the second derivative test can be a supplementary tool when analyzing the critical points you've found.