Problem 9 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: To determine the absolute maximum and minimum values of a continuous function on a closed interval, follow these steps: 1. Find all critical points of the function within the interval by taking the derivative of the function and solving for points where the derivative is equal to zero or undefined. 2. Evaluate the function at the critical points and endpoints of the interval, and make a list of these function values. 3. Compare the function values obtained in the previous step. The highest value is the absolute maximum, and the lowest value is the absolute minimum of the function on the closed interval.

Step 1: Find critical points

To find the critical points of a continuous function, we need to take its derivative and then find the points where the derivative is equal to zero or undefined. Let the given function be f(x). So, find f'(x) and solve for x such that f'(x) = 0 or f'(x) is undefined.

Step 2: Evaluate the function at critical points and endpoints

Evaluate the function f(x) at each critical point found in Step 1 and at the endpoints of the closed interval. Make a list of these function values.

Step 3: Compare function values to find absolute maximum and minimum

Compare the function values obtained in Step 2. The highest value is the absolute maximum, and the lowest value is the absolute minimum of the function on the closed interval.