/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 What does it mean for a function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 1

What does it mean for a function to have an absolute extreme value at a point \(c\) of an interval \([a, b] ?\)

Short Answer

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Question: Explain the concept of absolute extreme values of a function at a point within a closed interval and describe the closed interval method used to find them. Answer: Absolute extreme values are the highest (maximum) or lowest (minimum) values that a function reaches within a certain interval, also known as global maxima and global minima. A function has an absolute extreme value within the interval [a, b] at a point c if f(c) is either the highest or lowest value reached within the interval. A continuous function over a closed interval is necessary for determining absolute extreme values. The closed interval method is used to find the absolute extreme values of a continuous function over a closed interval [a, b]. This method involves four steps: 1. Find the critical points of the function by setting the derivative f'(x) equal to zero or determining where it is undefined. 2. Evaluate the function at each critical point found in step 1. 3. Evaluate the function at the endpoints of the interval, a and b. 4. Compare the function values from steps 2 and 3 to determine the highest and lowest values, which are the absolute maximum and minimum of the function within the interval. In summary, the absolute extreme values of a continuous function on a closed interval occur either at the critical points or the interval endpoints.

Step by step solution

01

Definition of Absolute Extreme Values

An absolute extreme value is the highest(maximum) or lowest(minimum) value that a function reaches within a certain interval. These are also known as global maxima and global minima.
02

Condition for a Function to Have an Absolute Extreme Value at a Point \(c\)

A function \(f(x)\) has an absolute extreme value at a point \(c\) within the interval \([a, b]\) if the value of \(f(c)\) is greater or equal to the value of \(f(x)\) for every point \(x\) in the interval (minimum), or if the value of \(f(c)\) is less than or equal to the value of \(f(x)\) for every point \(x\) in the interval (maximum). In simpler terms, \(f(c)\) must be either the highest or lowest value reached within the interval \([a, b]\).
03

Relationship with the Closed Interval

For a function to have an absolute extreme value at a point \(c\) within the interval \([a, b]\), the function must be continuous over the interval. This is because discontinuous functions may have jumps or gaps that make it difficult to determine whether a point is the highest or lowest value within the interval.
04

Closed Interval Method

To find the absolute extreme values of a continuous function \(f(x)\) over a closed interval \([a, b]\), we should follow these steps: 1. Find the critical points of the function by setting the derivative \(f'(x)\) equal to zero or determining where it is undefined. 2. Evaluate the function at each critical point found in step 1. 3. Evaluate the function at the endpoints of the interval, \(a\) and \(b\). 4. Compare the function values from steps 2 and 3. The highest value is the absolute maximum, while the lowest value is the absolute minimum. The main idea here is that the absolute extreme values of a continuous function on a closed interval occur either at the critical points or the interval endpoints.

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