/*! 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 5 Let \(\mathbf{f}\) be a function... [FREE SOLUTION] | 91影视

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Chapter 32: Problem 5

Let \(\mathbf{f}\) be a function of bounded variation on \([a, b]\) such that $$ f(x)>c>0 $$ Prove that \(1 / f\) is also a function of bounded variation and $$ V_{a}^{o}\left(\frac{1}{f}\right)<\frac{1}{c^{2}} V_{a}^{b}(f) $$

Short Answer

Expert verified
The function \(1/f\) is of bounded variation with \(V_{a}^{b}(1/f) < \frac{1}{c^2} V_{a}^{b}(f)\).

Step by step solution

01

Establish Definitions

First, recall the definitions involved in this problem: A function \( f \) is of bounded variation on \([a, b]\) if there exists a finite value \( V_a^b(f) \), the total variation. For a partition \( P = \{ x_0, x_1, \dots, x_n \} \) of \([a, b]\), the total variation is defined as \( V_a^b(f) = \sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| \). Given \( f(x) > c > 0 \) for all \( x \in [a, b] \), we aim to prove the bounded variation of \( \frac{1}{f} \).
02

Consider the Function 1/f

For each pair \((x_{i-1}, x_{i})\), consider the change in \( \frac{1}{f} \). We have: \[\frac{1}{f(x_i)} - \frac{1}{f(x_{i-1})} = \frac{f(x_{i-1}) - f(x_i)}{f(x_i)f(x_{i-1})}\]
03

Establish the Bound

By the given \( f(x) > c > 0 \), it follows \( f(x_{i})f(x_{i-1}) > c^2 \). Thus, \[ \left|\frac{1}{f(x_i)} - \frac{1}{f(x_{i-1})}\right| = \left|\frac{f(x_{i-1}) - f(x_i)}{f(x_i)f(x_{i-1})}\right| < \frac{|f(x_i) - f(x_{i-1})|}{c^2}\]
04

Sum The Inequalities

Sum the inequalities over the partition \( P \): \[V_a^b\left(\frac{1}{f}\right) = \sum_{i=1}^{n} \left| \frac{1}{f(x_i)} - \frac{1}{f(x_{i-1})} \right| < \frac{1}{c^2} \sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| = \frac{1}{c^2} V_a^b(f)\]
05

Conclusion

The inequality shows that the total variation \( V_a^b\left(\frac{1}{f}\right) \) is bounded by \( \frac{1}{c^2} V_a^b(f) \). Therefore, \( \frac{1}{f} \) is also a function of bounded variation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Total Variation
Total variation is a concept that helps us measure how much a function changes over a certain interval. Imagine you have a wavy line drawn on a graph between two points, say from point A to point B. The total variation is like asking how much the line wiggles, goes up, and down across this interval.
It's given the symbol \( V_a^b(f) \) where "a" and "b" represent the start and end points of your interval.
  • The total variation of a function \( f \) on \([a, b]\) is the sum of the absolute differences between the values at consecutive points within the interval.
  • A function is said to be of bounded variation on an interval if this total variation is less than some finite number.
The importance of total variation in this context lies in its ability to show that a function isn't wildly erratic or wildly fluctuating鈥攊t maintains some form of consistency across the interval. This concept is essential in function analysis, as it helps us quantify changes in various types of functions.
Insight into Function Analysis
Function analysis is a big part of mathematics that revolves around studying properties of functions, such as continuity, limits, and variations. It's like being a detective for math curves and equations. Here's why it's important:
  • Function analysis helps us understand how a function behaves over its domain鈥攚hether it increases, decreases, or remains constant.
  • It also examines changes and rates of change within the functions.
  • Through function analysis, we can predict future behavior and establish relationships between various mathematical expressions.
When a function is said to be of bounded variation, as we're discussing here, function analysis assists in identifying how much the function's output varies over a given distance. This is crucial because it can determine how smooth or jagged a function might appear鈥攌ey information for applications ranging from engineering to economics.
Exploring Inequality in Mathematics
Inequality in mathematics refers to a relationship between two values that are not equal, and it's a very helpful concept for comparing quantities. You often see symbols like \( <, >, \leq, \) and \( \geq \) in inequalities.
  • Inequalities allow us to define boundaries or limits for a function or a value.
  • They can express conditions under which one function will always produce a value greater than, less than, or equal to another function.
  • In the context of this problem, an inequality helps us see that \( 1/f \) has a total variation that is less than some factor of the total variation of \( f \).
Understanding inequalities helps create a visual range for a function's possible behavior, letting us know "how much" one value can differ from the other. When dealing with functions, knowing this can aid significantly in predicting outcomes and forming solutions to complex mathematical challenges.

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Most popular questions from this chapter

Prove that the function $$ f(x)= \begin{cases}x^{\alpha} \sin \frac{1}{x^{\beta}} & \text { if } \quad 0\beta\) but not if \(\alpha \leqslant \beta\).

Suppose \(\mathbf{f}\) has a bounded derivative on \([\mathrm{a}, \mathrm{b}]\), so that \(f^{\prime}(x)\) exists and satisfies an inequality \(\left|f^{\prime}(x)\right|

Let \(\mathbf{f}\) be a function of bounded variation on \([\mathrm{a}, \mathrm{b}]\). Prove that $$ \|f\|=V_{a}^{b}(f) $$ has all the properties of a norm (cf. p. 138 ) if we impose the extra condition \(\mathbf{f}(\mathrm{a})=0\) Comment. Thus the space \(V_{[a, b]}^{0}\) of all functions of bounded variation on \([a, b]\) equipped with this norm and vanishing at \(x=a\) is a normed linear space (addition of functions and multiplication of functions by numbers being defined in the usual way).
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