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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7, Problem 78RE
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using Lagrange Multipliers In Exercises 73-78, use Lagrange multipliers to find the indicated extremum. Assume that the variables are positive.

Maximize f ( x ,   y , z ) = x 2 z   +   y z

Constraint: 2 x   +   y   +   z =   5

To determine

To calculate: The maximum value of the function f(x,y,z)=x2z+yz subject to constraint 2x+y+z=5 by the application of Lagrange multipliers.

Explanation

Given Information:

The provided function is f(x,y,z)=x2z+yz and the subject to constraints are 2x+y+z=5.

Formula used:

To find the maximum or minimum values of function f follow the steps as,

Step 1: Solve the provided system of equations as;

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Step 2: Next calculate the function f at each of the solution point obtained from first step where the greatest value obtained gives the maximum of function f subject to constraint g(x,y)=0 and the least value gives the minimum of function f subject to constraint g(x,y)=0.

Calculation:

First let f(x,y,z)=x2z+yz and g(x,y,z)=2x+y+z−5.

Then define a new function F as;

F(x,y,z,λ)=f(x,y,z)−λg(x,y,z)=x2z+yz−λ(2x+y+z−5)

Now solve the equation by finding the partial derivatives of F with respect to x, y, z and λ as;

Fx(x,y,z,λ)=2xz−2λFy(x,y,z,λ)=z−λFz(x,y,z,λ)=x2+y−λFλ(x,y,z,λ)=−(2x+y+z−5)

Then set each of the obtained partial derivatives equal to 0 as;

2xz−2λ=0z−λ=0x2+y−λ=0&

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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Radicals and Exponents Simplify the expression. 17. (a) (a2)3(a3b)2(b3)4 (b) (3xy2)3(23x1y)2

Precalculus: Mathematics for Calculus (Standalone Book)

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to ...

Multivariable Calculus

A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is o...

Single Variable Calculus: Early Transcendentals, Volume I

SOC Data on several variables measuring overall health and well-being for 11 nations are reported here for 2010...

Essentials Of Statistics

Place the following scores in a frequency distribution table. Based on the frequencies, what is the shape of th...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Trace or copy the graph of the given function f. Assume that the axes have equal scales. Then use the method of...

Calculus (MindTap Course List)

Global Defense Spending Global defense spending stood at 1.44 trillion in 2009 and is projected to grow at the ...

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

Interpretation Lucy conducted a survey asking some of her friends to specify their favorite type of TV entertai...

Understanding Basic Statistics

A set of scores has been organized into the following stem and leaf display. For this set of scores: a. How man...

Statistics for The Behavioral Sciences (MindTap Course List)

Find the domain and range and sketch the graph of the function h(x)=4x2.

Single Variable Calculus

In Exercises 65 to 68, draw a Venn diagram with each of the given element placed in the correct region. U = {2,...

Mathematical Excursions (MindTap Course List)

In problems 31-36, simplify each expression. 36.

Mathematical Applications for the Management, Life, and Social Sciences

A transmitter is sending a message by using a binary code, namely, a sequence of 0s and 1s. Each transmitted bi...

Probability and Statistics for Engineering and the Sciences

In Exercises 3148, (a) factor the given expression, and (b) set the expression equal to zero and solve for the ...

Finite Mathematics

For week 1 of a week-loss competition, Team A had three members lose 8 pounds each, two members lose 5 pounds e...

Intermediate Algebra

Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at t...

Single Variable Calculus: Early Transcendentals

A boat is 6 m below the level of a pier and 12 m from the pier as measured across the water. How much rope is n...

Elementary Geometry For College Students, 7e

Prior to an election, a political research firm announced that a candidate for mayor had gained 8 percentage po...

Contemporary Mathematics for Business & Consumers

Prove that if and are integers such that and , then .

Elements Of Modern Algebra

Add: (8)+(3)

Elementary Technical Mathematics

Prove that each of the following identities is true. sin2x(cot2x+1)=1

Trigonometry (MindTap Course List)

Finding a Limit In Exercises 5564, find the limit, if it exists. limx3x2x+5

Calculus of a Single Variable

Finding an Inverse Function In Exercises 63-68, (a) sketch a graph of the function f, (b) determine an interval...

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercises 2932, a fair die is dolled, and the number that falls uppermost is observed. Let E and F be events...

Finite Mathematics for the Managerial, Life, and Social Sciences

The Angle-Addition Postulate can be generalized as follows: The measure of an angle equals the sum of measures ...

Elementary Geometry for College Students

MixtureIn Exercises 3538, consider a tank that at time t=0 contains v0 gallons of a solution of which, by weigh...

Calculus

For j – 2k 3i – 4k 3i + j – 2k 0 (zero vector)

Study Guide for Stewart's Multivariable Calculus, 8th

limx2x2(x1)1x = _____. a) 4 b) 1 c) 0 d) does not exist

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

ProofIf r(t) is a nonzero differentiable function of t, prove that ddt[r(t)]=r(t)r(t)r(t)

Multivariable Calculus

Matching In Exercises 5-8, match the Taylor polynomial approximation of the function f(x)=ex2/2 with its graph....

Calculus (MindTap Course List)

Write each inequality in interval notation and graph the interval. x5

College Algebra (MindTap Course List)

Describe the four probability sampling methods presented in the book, other than simple random sampling (strati...

Research Methods for the Behavioral Sciences (MindTap Course List)

A data set has a first quartile of 42 and a third quartile of 50. Compute the lower and upper limits for the co...

Statistics for Business & Economics, Revised (MindTap Course List)

Explain why plagiarism is unethical.

Research Methods for the Behavioral Sciences (MindTap Course List)

A researcher at the Medical College of Virginia conducted a study of 60 randomly selected male soccer players a...

Introduction To Statistics And Data Analysis

Given the number of cuts, the length of cut, the revolutions per minute of the workpiece or tool, and the tool ...

Mathematics For Machine Technology

A trucking firm handles four commodities for a particular distributor. Total shipments for the commodities in 2...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

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