Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 832

If two variables \(x\) and \(y\) and \(x>0 . x y=1\) then minimum value of \(\mathrm{x}+\mathrm{y}\) is (a) 1 (b) 2 (c) \(2(1 / 2)\) (d) \(3(1 / 3)\)

Short Answer

Expert verified
The minimum value of \(x+y\) under the given constraints is (b) 2.

Step by step solution

01

Write down given constraints and expression to minimize

We are given the constraints x > 0 and xy = 1, and we want to find the minimum value of x + y.
02

Apply AM-GM inequality to x and y

The Arithmetic Mean (AM) of x and y is given by \(\frac{x+y}{2}\). The Geometric Mean (GM) of x and y is given by \(\sqrt{xy}\). According to AM-GM inequality, we have: \[\frac{x+y}{2} \geq \sqrt{xy}\]
03

Use the given constraint xy = 1

Since we have xy = 1, we can substitute this value into the inequality from Step 2: \[\frac{x+y}{2} \geq \sqrt{1}\]
04

Simplify the inequality and solve for x + y

The inequality from Step 3 simplifies to: \[\frac{x+y}{2} \geq 1\] Now, multiply both sides of the inequality by 2 to isolate x + y: \[x+y \geq 2\]
05

Identify the minimum value of x + y

From the inequality \(x+y \geq 2\), we can conclude that the minimum value of x + y is 2. So, the correct answer is: (b) 2

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \(\mathrm{f}(\mathrm{x})=[\mathrm{x} /(\sin \mathrm{x})]\) and \(\mathrm{g}(\mathrm{x})=[\mathrm{x} /(\tan \mathrm{x})]\) where \(0<\mathrm{x} \leq 1\) than in this interval (a) both \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are increasing function (b) both \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are decreasing function (c) \(\mathrm{f}(\mathrm{x})\) is an increasing function (d) \(\mathrm{g}(\mathrm{x})\) is an increasing function

For \((\mathrm{d} / \mathrm{dx}) \log \left|\mathrm{e}^{\mathrm{x}}[(\mathrm{x}-4) /(\mathrm{x}+4)]^{(3 / 4)}\right|\) at \(\mathrm{x}=5\) then \((\mathrm{dy} / \mathrm{d} \mathrm{x})\) \(\begin{array}{llll}\text { (a) }(5 / 3) & \text { (b) }(3 / 5) & \text { (c) }-(3 / 5) & \text { (d) }-(5 / 3)\end{array}\)

If \(\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})=\mathrm{f}^{\prime}(\mathrm{x})\) and \(\mathrm{F}(\mathrm{x})=|[\mathrm{f}(\mathrm{x} / 2)]|^{2}+|[\mathrm{g}(\mathrm{x} / 2)]|^{2}\) and \(F(5)=5\) then \(F(10)=\) (a) 5 (b) 10 (c) (d) 15

If the curves \(2 \mathrm{x}^{2}+3 \mathrm{y}^{2}=6\) and \(\mathrm{ax}^{2}+4 \mathrm{y}^{2}=4\) intersect orthogonally than a = (a) 2 (b) 1 (c) 3 (d) \(-3\)

\(\mathrm{f}(\mathrm{x})=\mathrm{x} \cdot \sin (1 / \mathrm{x})\) and \(\mathrm{x} \in[-1,1]\). Also \(\mathrm{f}(0)=0\) then. (a) \(\mathrm{f}(\mathrm{x})\) is continuous in \([-1,1]\) (b) Roll's theorem is applicable in \([-1,1]\) (c) First mean value theorem is applicable in \([-1,1]\) (d) none of these
See all solutions

Recommended explanations on English Textbooks

Discourse

Read Explanation

Essay Prompts

Read Explanation

Global English

Read Explanation

English Grammar Summary

Read Explanation

Graphology

Read Explanation

Language and Social Groups

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.

Sign-up for free