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Chapter 4: Problem 9

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

Short Answer

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Question: Given a continuous function f(x) on a closed interval [a, b], find the absolute maximum and minimum values of the function. Answer: To find the absolute maximum and minimum values of the function, follow these steps: 1. Identify the critical points by finding the derivative (f'(x)) and setting it to zero. Solve the equation f'(x) = 0 for x to find the critical points. 2. Evaluate the function f(x) at the critical points obtained in Step 1. 3. Evaluate the function f(x) at the endpoints of the interval (a and b). 4. Compare the function values obtained in Steps 2 and 3 to determine the highest (absolute maximum) and lowest (absolute minimum) values. 5. Report the absolute maximum and minimum values, along with their corresponding x-coordinates where these values occur.

Step by step solution

01

Identify the critical points

To find the critical points, differentiate the function with respect to the variable and set the derivative equal to zero. The points where the derivative is equal to zero or where the derivative is undefined are the critical points. Let's denote the function as f(x) and its derivative as f'(x). Solve the equation f'(x) = 0 for x, and these will be the critical points.
02

Evaluate the function at the critical points

Now that you have found the critical points, evaluate the function f(x) at each of the critical points. These values will help determine if there's a maximum or minimum value for the function on the given interval.
03

Evaluate the function at the endpoints of the interval

In order to consider all possible maximum and minimum values of the function, we also need to evaluate f(x) at the endpoints of the interval. Let's say the given interval is [a, b]. Calculate the values of f(a) and f(b).
04

Compare the function values obtained in Steps 2 and 3

By comparing the function values obtained in Steps 2 and 3, we can identify the highest and lowest values among them. The highest function value represents the absolute maximum value of the function on the given interval, and the lowest function value represents the absolute minimum value of the function on the interval.
05

Report the absolute maximum and minimum values

Finally, report the absolute maximum and minimum values of the function on the given closed interval, along with their corresponding points (x-coordinates) where these values occur.

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