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Chapter 4: Problem 3

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

Short Answer

Expert verified
Answer: The conditions necessary for a function to have an absolute maximum and an absolute minimum value on an interval are: 1. The function must be continuous on the interval. 2. The interval must be closed [a, b]. If both of these conditions are met, the function is guaranteed to have an absolute maximum and an absolute minimum value on that interval.

Step by step solution

01

Understanding Absolute Maximum and Minimum

An absolute maximum value for a function is the highest output value (function value) that the function can have within an interval. Similarly, an absolute minimum value for a function is the lowest output value that the function can have within an interval. To understand this better, let's consider a function f(x) defined on an interval [a, b]. In this case, the absolute maximum value of f(x) would be the largest f(x) value for x in [a, b], and the absolute minimum value of f(x) would be the smallest f(x) value for x in [a, b].
02

Roles of Intervals in Absolute Maximum and Minimum

An interval can be open (a, b), closed [a, b], or semi-open (a, b] or [a, b). Absolute maximum and minimum values are most commonly found in closed intervals as they contain a limited and well-defined range of values for x. However, open and semi-open intervals could also have absolute maximum and minimum values if certain criteria are met.
03

The Extreme Value Theorem

The Extreme Value Theorem (EVT), a fundamental result in calculus, guarantees the existence of absolute maximum and minimum values for a function defined on a closed interval [a, b]. According to EVT, if the function is continuous on [a, b], then the function will have at least one absolute maximum value and at least one absolute minimum value in the interval.
04

Conditions for Absolute Maximum and Minimum

Based on the Extreme Value Theorem and our understanding of intervals, we can summarize the conditions that must be met to ensure a function has an absolute maximum value and an absolute minimum value on an interval: 1. The function must be continuous on the interval. 2. The interval must be closed [a, b]. If both of these conditions are met, then the function is guaranteed to have an absolute maximum and an absolute minimum value on that interval. Note that these conditions are sufficient but not necessary, as there could be cases where a function has an absolute maximum and minimum, even if the interval is not closed or the function is not continuous throughout the interval.

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