/*! 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 107 Use the definitions of increasin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 107

Use the definitions of increasing and decreasing functions to prove that \(f(x)=1 / x\) is decreasing on \((0, \infty)\).

Short Answer

Expert verified
By choosing any two arbitrary points 'a' and 'b' within the domain (0, \(\infty\)) and such that \(a < b\), and by evaluating the function at these points, it can be observed that \(f(a) > f(b)\). Hence, by definition, the function \(f(x) = 1 / x\) is decreasing for \(x > 0\).

Step by step solution

01

Define the function and its domain

The function to be tested is \(f(x) = 1 / x\) and the domain that we need to consider is \(x\) greater than 0 (i.e., (0, \(\infty\))).
02

Choose arbitrary numbers within the domain

Select arbitrary numbers \'a\' and \'b\' in the domain such that \(0 < a < b\). Now, the task is to prove that whenever \(a < b\), \(f(a) > f(b)\). Let's compute the function values at these points and compare them.
03

Compute function values at chosen points

The value of the function \(f(x)\) at points \'a\' and \'b\' are \(f(a) = 1/a\) and \(f(b) = 1/b\) respectively.
04

Comparison of function values

Since a and b are positive and \(a < b\), it implies that \(1/a > 1/b\). Thus, it can be said that \(f(a) > f(b)\) whenever \(a < b\). Therefore, by definition, the function \(f(x) = 1 / x\) is decreasing on the domain (0, \(\infty\)).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{3}}{\sqrt{x^{2}-4}} $$

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\sin \left(\frac{x}{x-2}\right), \quad x>3 $$

Consider \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(N\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{2}}{x^{2}+9} $$

Timber Yield The yield \(V\) (in millions of cubic feet per acre) for a stand of timber at age \(t\) (in years) is \(V=7.1 e^{(-48.1) / t}\) (a) Find the limiting volume of wood per acre as \(t\) approaches infinity. (b) Find the rates at which the yield is changing when \(t=20\) years and \(t=60\) years.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.