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Chapter 4: Problem 5

How do you find the absolute maximum and minimum values of a function that is continuous on a closed interval?

Short Answer

Expert verified
Answer: To find the absolute maximum and minimum values of a continuous function on a closed interval, follow these steps: 1. Determine the critical points of the function by finding the derivative and setting it equal to zero. 2. Calculate the function's values at the critical points and endpoints of the interval. 3. Compare the calculated values to determine the absolute maximum and minimum values.

Step by step solution

01

Determine Critical Points

First, find the first derivative, f'(x), of the function f(x). This will identify the points where the function's slope changes, indicating a possible maximum or minimum value. Then, set the derivative equal to zero and solve for x to find the critical points.
02

Calculate Function Values at Critical Points and Endpoints

Once you have found the critical points, evaluate the function at these critical points and the endpoints of the interval. This can be done by plugging each value into the original function, f(x), and calculating the corresponding function values.
03

Compare Function Values

Now that you have the function values for all of the critical points and endpoints, you can compare them to determine which values represent the absolute maximum and minimum function values on the given interval. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum. Keep in mind that there may be more than one point representing the absolute maximum or minimum value.

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