BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Solutions

Chapter
Section
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Explain how the method of Lagrange multipliers works in finding the extreme values of f(x, y, z) subject to the constraint g(x, y, z) = k. What if there is a second constraint h(x, y, z) = c?

To determine

To explain: The method of Lagrange multipliers for finding the extreme values f(x,y,z) subject to the constraint g(x,y,z)=k ; what happens if there is a second constraint h(x,y,z)=c .

Explanation

Step 1:

Consider the function f(x,y,z) .

Find the gradient vector of the function f(x,y,z) .

f(x,y,z)=fxi+fxj+fxk=fx,fy,fz

Find the gradient vector of the function g(x,y,z)=k .

g(x,y,z)=gxi+gxj+gxk=gx,gy,gz

Step 2:

Comparing the following equations as follows,

f(x,y,z)=λg(x,y,z)

fx=λgx (1)

fy=λgy (2)

fz=λgz (3)

g(x,y,z)=k (4)

Step 3:

From the equations (1),(2),(3) and (4),

Compute the values of x,yandz .

Substitute the values of x,yandz in the equation f(x,y,z) and obtain the absolute maximum and absolute minimum values.

This is the required solution.

If there are two constraints follow the below method.

Step 1:

Consider the function f(x,y,z) .

Find the gradient vector of the function f(x,y,z) .

f(x,y,z)=fxi+fxj+fxk=fx,fy,fz

Find the gradient vector of the function g(x,y,z)=k

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 14 Solutions

Show all chapter solutions add
Sect-14.1 P-11ESect-14.1 P-12ESect-14.1 P-13ESect-14.1 P-14ESect-14.1 P-15ESect-14.1 P-16ESect-14.1 P-17ESect-14.1 P-18ESect-14.1 P-19ESect-14.1 P-20ESect-14.1 P-21ESect-14.1 P-22ESect-14.1 P-23ESect-14.1 P-24ESect-14.1 P-25ESect-14.1 P-26ESect-14.1 P-27ESect-14.1 P-28ESect-14.1 P-29ESect-14.1 P-30ESect-14.1 P-31ESect-14.1 P-32ESect-14.1 P-33ESect-14.1 P-34ESect-14.1 P-35ESect-14.1 P-36ESect-14.1 P-37ESect-14.1 P-38ESect-14.1 P-39ESect-14.1 P-40ESect-14.1 P-41ESect-14.1 P-42ESect-14.1 P-43ESect-14.1 P-44ESect-14.1 P-45ESect-14.1 P-46ESect-14.1 P-47ESect-14.1 P-48ESect-14.1 P-49ESect-14.1 P-50ESect-14.1 P-51ESect-14.1 P-52ESect-14.1 P-53ESect-14.1 P-54ESect-14.1 P-55ESect-14.1 P-56ESect-14.1 P-57ESect-14.1 P-58ESect-14.1 P-59ESect-14.1 P-60ESect-14.1 P-61ESect-14.1 P-62ESect-14.1 P-63ESect-14.1 P-64ESect-14.1 P-65ESect-14.1 P-66ESect-14.1 P-67ESect-14.1 P-68ESect-14.1 P-69ESect-14.1 P-70ESect-14.1 P-71ESect-14.1 P-72ESect-14.1 P-73ESect-14.1 P-74ESect-14.1 P-75ESect-14.1 P-76ESect-14.1 P-77ESect-14.1 P-78ESect-14.1 P-79ESect-14.1 P-80ESect-14.1 P-81ESect-14.2 P-1ESect-14.2 P-2ESect-14.2 P-3ESect-14.2 P-4ESect-14.2 P-5ESect-14.2 P-6ESect-14.2 P-7ESect-14.2 P-8ESect-14.2 P-9ESect-14.2 P-10ESect-14.2 P-11ESect-14.2 P-12ESect-14.2 P-13ESect-14.2 P-14ESect-14.2 P-15ESect-14.2 P-16ESect-14.2 P-17ESect-14.2 P-18ESect-14.2 P-19ESect-14.2 P-20ESect-14.2 P-21ESect-14.2 P-22ESect-14.2 P-23ESect-14.2 P-24ESect-14.2 P-25ESect-14.2 P-26ESect-14.2 P-27ESect-14.2 P-28ESect-14.2 P-29ESect-14.2 P-30ESect-14.2 P-31ESect-14.2 P-32ESect-14.2 P-33ESect-14.2 P-34ESect-14.2 P-35ESect-14.2 P-36ESect-14.2 P-37ESect-14.2 P-38ESect-14.2 P-39ESect-14.2 P-40ESect-14.2 P-41ESect-14.2 P-42ESect-14.2 P-43ESect-14.2 P-44ESect-14.2 P-45ESect-14.2 P-46ESect-14.3 P-1ESect-14.3 P-2ESect-14.3 P-3ESect-14.3 P-4ESect-14.3 P-5ESect-14.3 P-6ESect-14.3 P-7ESect-14.3 P-8ESect-14.3 P-9ESect-14.3 P-10ESect-14.3 P-11ESect-14.3 P-12ESect-14.3 P-13ESect-14.3 P-14ESect-14.3 P-15ESect-14.3 P-16ESect-14.3 P-17ESect-14.3 P-18ESect-14.3 P-19ESect-14.3 P-20ESect-14.3 P-21ESect-14.3 P-22ESect-14.3 P-23ESect-14.3 P-24ESect-14.3 P-25ESect-14.3 P-26ESect-14.3 P-27ESect-14.3 P-28ESect-14.3 P-29ESect-14.3 P-30ESect-14.3 P-31ESect-14.3 P-32ESect-14.3 P-33ESect-14.3 P-34ESect-14.3 P-35ESect-14.3 P-36ESect-14.3 P-37ESect-14.3 P-38ESect-14.3 P-39ESect-14.3 P-40ESect-14.3 P-41ESect-14.3 P-42ESect-14.3 P-43ESect-14.3 P-44ESect-14.3 P-45ESect-14.3 P-46ESect-14.3 P-47ESect-14.3 P-48ESect-14.3 P-49ESect-14.3 P-50ESect-14.3 P-51ESect-14.3 P-52ESect-14.3 P-53ESect-14.3 P-54ESect-14.3 P-55ESect-14.3 P-56ESect-14.3 P-57ESect-14.3 P-58ESect-14.3 P-59ESect-14.3 P-60ESect-14.3 P-61ESect-14.3 P-62ESect-14.3 P-63ESect-14.3 P-64ESect-14.3 P-65ESect-14.3 P-66ESect-14.3 P-67ESect-14.3 P-68ESect-14.3 P-69ESect-14.3 P-70ESect-14.3 P-71ESect-14.3 P-72ESect-14.3 P-73ESect-14.3 P-74ESect-14.3 P-75ESect-14.3 P-76ESect-14.3 P-77ESect-14.3 P-78ESect-14.3 P-79ESect-14.3 P-80ESect-14.3 P-81ESect-14.3 P-82ESect-14.3 P-83ESect-14.3 P-84ESect-14.3 P-85ESect-14.3 P-86ESect-14.3 P-87ESect-14.3 P-88ESect-14.3 P-89ESect-14.3 P-90ESect-14.3 P-91ESect-14.3 P-92ESect-14.3 P-93ESect-14.3 P-94ESect-14.3 P-95ESect-14.3 P-96ESect-14.3 P-97ESect-14.3 P-98ESect-14.3 P-99ESect-14.3 P-100ESect-14.3 P-101ESect-14.3 P-102ESect-14.3 P-103ESect-14.3 P-104ESect-14.4 P-1ESect-14.4 P-2ESect-14.4 P-3ESect-14.4 P-4ESect-14.4 P-5ESect-14.4 P-6ESect-14.4 P-7ESect-14.4 P-8ESect-14.4 P-11ESect-14.4 P-12ESect-14.4 P-13ESect-14.4 P-14ESect-14.4 P-15ESect-14.4 P-16ESect-14.4 P-17ESect-14.4 P-18ESect-14.4 P-19ESect-14.4 P-20ESect-14.4 P-21ESect-14.4 P-22ESect-14.4 P-23ESect-14.4 P-24ESect-14.4 P-25ESect-14.4 P-26ESect-14.4 P-27ESect-14.4 P-28ESect-14.4 P-29ESect-14.4 P-30ESect-14.4 P-31ESect-14.4 P-32ESect-14.4 P-33ESect-14.4 P-34ESect-14.4 P-35ESect-14.4 P-36ESect-14.4 P-37ESect-14.4 P-38ESect-14.4 P-39ESect-14.4 P-40ESect-14.4 P-41ESect-14.4 P-42ESect-14.4 P-43ESect-14.4 P-44ESect-14.4 P-45ESect-14.4 P-46ESect-14.5 P-1ESect-14.5 P-2ESect-14.5 P-3ESect-14.5 P-4ESect-14.5 P-5ESect-14.5 P-6ESect-14.5 P-7ESect-14.5 P-8ESect-14.5 P-9ESect-14.5 P-10ESect-14.5 P-11ESect-14.5 P-12ESect-14.5 P-13ESect-14.5 P-14ESect-14.5 P-15ESect-14.5 P-16ESect-14.5 P-17ESect-14.5 P-18ESect-14.5 P-19ESect-14.5 P-20ESect-14.5 P-21ESect-14.5 P-22ESect-14.5 P-23ESect-14.5 P-24ESect-14.5 P-25ESect-14.5 P-26ESect-14.5 P-27ESect-14.5 P-28ESect-14.5 P-29ESect-14.5 P-30ESect-14.5 P-31ESect-14.5 P-32ESect-14.5 P-33ESect-14.5 P-34ESect-14.5 P-35ESect-14.5 P-36ESect-14.5 P-37ESect-14.5 P-38ESect-14.5 P-39ESect-14.5 P-40ESect-14.5 P-41ESect-14.5 P-42ESect-14.5 P-43ESect-14.5 P-44ESect-14.5 P-45ESect-14.5 P-46ESect-14.5 P-47ESect-14.5 P-48ESect-14.5 P-49ESect-14.5 P-50ESect-14.5 P-51ESect-14.5 P-53ESect-14.5 P-55ESect-14.5 P-56ESect-14.5 P-57ESect-14.5 P-58ESect-14.5 P-59ESect-14.6 P-1ESect-14.6 P-2ESect-14.6 P-3ESect-14.6 P-4ESect-14.6 P-5ESect-14.6 P-6ESect-14.6 P-7ESect-14.6 P-8ESect-14.6 P-9ESect-14.6 P-10ESect-14.6 P-11ESect-14.6 P-12ESect-14.6 P-13ESect-14.6 P-14ESect-14.6 P-15ESect-14.6 P-16ESect-14.6 P-17ESect-14.6 P-18ESect-14.6 P-19ESect-14.6 P-20ESect-14.6 P-21ESect-14.6 P-22ESect-14.6 P-23ESect-14.6 P-24ESect-14.6 P-25ESect-14.6 P-26ESect-14.6 P-27ESect-14.6 P-28ESect-14.6 P-29ESect-14.6 P-30ESect-14.6 P-31ESect-14.6 P-32ESect-14.6 P-33ESect-14.6 P-34ESect-14.6 P-35ESect-14.6 P-36ESect-14.6 P-37ESect-14.6 P-38ESect-14.6 P-39ESect-14.6 P-40ESect-14.6 P-41ESect-14.6 P-42ESect-14.6 P-43ESect-14.6 P-44ESect-14.6 P-45ESect-14.6 P-46ESect-14.6 P-47ESect-14.6 P-48ESect-14.6 P-49ESect-14.6 P-50ESect-14.6 P-51ESect-14.6 P-52ESect-14.6 P-53ESect-14.6 P-54ESect-14.6 P-55ESect-14.6 P-56ESect-14.6 P-57ESect-14.6 P-58ESect-14.6 P-59ESect-14.6 P-60ESect-14.6 P-61ESect-14.6 P-62ESect-14.6 P-63ESect-14.6 P-64ESect-14.6 P-65ESect-14.6 P-66ESect-14.6 P-67ESect-14.6 P-68ESect-14.6 P-69ESect-14.6 P-70ESect-14.7 P-1ESect-14.7 P-2ESect-14.7 P-3ESect-14.7 P-4ESect-14.7 P-5ESect-14.7 P-6ESect-14.7 P-7ESect-14.7 P-8ESect-14.7 P-9ESect-14.7 P-10ESect-14.7 P-11ESect-14.7 P-12ESect-14.7 P-13ESect-14.7 P-14ESect-14.7 P-15ESect-14.7 P-16ESect-14.7 P-17ESect-14.7 P-18ESect-14.7 P-19ESect-14.7 P-20ESect-14.7 P-21ESect-14.7 P-22ESect-14.7 P-23ESect-14.7 P-24ESect-14.7 P-25ESect-14.7 P-26ESect-14.7 P-27ESect-14.7 P-28ESect-14.7 P-29ESect-14.7 P-30ESect-14.7 P-31ESect-14.7 P-32ESect-14.7 P-33ESect-14.7 P-34ESect-14.7 P-35ESect-14.7 P-36ESect-14.7 P-37ESect-14.7 P-38ESect-14.7 P-39ESect-14.7 P-40ESect-14.7 P-41ESect-14.7 P-42ESect-14.7 P-43ESect-14.7 P-44ESect-14.7 P-45ESect-14.7 P-46ESect-14.7 P-47ESect-14.7 P-48ESect-14.7 P-49ESect-14.7 P-50ESect-14.7 P-51ESect-14.7 P-52ESect-14.7 P-53ESect-14.7 P-54ESect-14.7 P-55ESect-14.7 P-56ESect-14.7 P-57ESect-14.7 P-58ESect-14.7 P-59ESect-14.7 P-60ESect-14.8 P-1ESect-14.8 P-2ESect-14.8 P-3ESect-14.8 P-4ESect-14.8 P-5ESect-14.8 P-6ESect-14.8 P-7ESect-14.8 P-8ESect-14.8 P-9ESect-14.8 P-10ESect-14.8 P-11ESect-14.8 P-12ESect-14.8 P-13ESect-14.8 P-14ESect-14.8 P-15ESect-14.8 P-16ESect-14.8 P-17ESect-14.8 P-18ESect-14.8 P-19ESect-14.8 P-20ESect-14.8 P-21ESect-14.8 P-22ESect-14.8 P-23ESect-14.8 P-24ESect-14.8 P-25ESect-14.8 P-27ESect-14.8 P-28ESect-14.8 P-29ESect-14.8 P-30ESect-14.8 P-31ESect-14.8 P-32ESect-14.8 P-33ESect-14.8 P-34ESect-14.8 P-35ESect-14.8 P-36ESect-14.8 P-37ESect-14.8 P-38ESect-14.8 P-39ESect-14.8 P-40ESect-14.8 P-41ESect-14.8 P-42ESect-14.8 P-43ESect-14.8 P-44ESect-14.8 P-45ESect-14.8 P-46ESect-14.8 P-49ESect-14.8 P-50ECh-14 P-1RCCCh-14 P-2RCCCh-14 P-3RCCCh-14 P-4RCCCh-14 P-5RCCCh-14 P-6RCCCh-14 P-7RCCCh-14 P-8RCCCh-14 P-9RCCCh-14 P-10RCCCh-14 P-11RCCCh-14 P-12RCCCh-14 P-13RCCCh-14 P-14RCCCh-14 P-15RCCCh-14 P-16RCCCh-14 P-17RCCCh-14 P-18RCCCh-14 P-19RCCCh-14 P-1RQCh-14 P-2RQCh-14 P-3RQCh-14 P-4RQCh-14 P-5RQCh-14 P-6RQCh-14 P-7RQCh-14 P-8RQCh-14 P-9RQCh-14 P-10RQCh-14 P-11RQCh-14 P-12RQCh-14 P-1RECh-14 P-2RECh-14 P-3RECh-14 P-4RECh-14 P-5RECh-14 P-6RECh-14 P-7RECh-14 P-8RECh-14 P-9RECh-14 P-10RECh-14 P-11RECh-14 P-12RECh-14 P-13RECh-14 P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Use Exercise 51 to find (lnx)3dx.

Calculus (MindTap Course List)

In Exercises 1-8, use the graph of the given function f to determine limxaf(x) at the indicated value of a, if ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Factor each expression in Problems 7-18 as a product of binomials. 7.

Mathematical Applications for the Management, Life, and Social Sciences

Factor each expression completely: 4x225

Elementary Technical Mathematics

True or False: converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th