Problem 3 - Calculus Early Transcendentals

Question

What conditions must be met to ensure that a function has an absolute maximum value and an absolute minimum value on an interval?

Step-by-Step Solution

Answer

Answer: For a function to have both an absolute maximum and absolute minimum value on an interval, the function must be continuous on the interval, and the interval must be a closed interval that includes both its endpoints. These conditions are guaranteed by the Extreme Value Theorem.

Step 1: Understand the Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then the function has both an absolute maximum value and an absolute minimum value on that interval. This means that there exist points within the interval where the function achieves its highest and lowest values.

Step 2: Identify the conditions for a function to be continuous

A function is continuous if it has no breaks, holes, or jumps along its domain. In other words, for any point x in its domain, the limit of the function as x approaches that point exists and is equal to the function's value at that point. In mathematical terms, if f(x) is a function: $$\lim_{x \to c} f(x) = f(c)$$ This must hold true for every point c in the domain of the function.

Step 3: Determine the conditions for a closed interval

A closed interval [a, b] is an interval that includes both its endpoints, a and b. This means that any value within the interval, including a and b, has a corresponding function value.

Step 4: Combine the conditions for the Extreme Value Theorem

To ensure that a function has an absolute maximum value and an absolute minimum value on an interval, the function must meet the following conditions: 1. The function must be continuous on the interval. 2. The interval must be a closed interval, including both its endpoints. By fulfilling these conditions, a function will possess both an absolute maximum and an absolute minimum value on the given interval, as guaranteed by the Extreme Value Theorem.