   Chapter 7.6, Problem 4E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Using Lagrange Multipliers In Exercises 1-12, use Lagrange multipliers to find the indicated extremum. Assume that x and y are positive. See Example 1.Maximize f ( x , y ) = x 2 + y 2 Constraint: − 2 x − 4 y = − 5

To determine

To calculate: The minimum value of the function f(x,y)=x2+y2 subject to the constraint 2x4y=5 by the use of Lagrange multipliers.

Explanation

Given Information:

The provided function is f(x,y)=x2+y2 subject to the constraint 2x4y=5. Consider that the variables x and y are positive.

Formula used:

Method of Lagrange multipliers,

If the function f(x,y) contains a maximum or minimum subject to the constraint g(x,y)=0 then the maximum or minimum can occur at one of the critical numbers of the function F is,

F(x,y,λ)=f(x,y)λg(x,y) where, λ is a Lagrange multiplier.

Steps to determine the minimum or maximum of the function f.

1. Solve the system of equations,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

2. Determine the value of the function f at each solution obtained from the step 1.

The largest value gives the maximum value of function f subject to the constraint g(x,y)=0 and the lowest value gives the minimum value of function f subject to the constraint g(x,y)=0.

Calculation:

Consider the function,

f(x,y)=x2+y2

The provided constraint is,

2x4y=52x4y+5=0

So, g(x,y)=2x4y+5

Now, the new function F is,

F(x,y,λ)=x2+y2λ(2x4y+5)

Find the partial derivative of function F to determine the critical number for F.

The partial derivatives of F with respect to x and then set it to zero.

Fx(x,y,λ)=x[x2+y2λ(2x4y+5)]0=2x+2λx=λ

The partial derivatives of F with respect to y and then set it to zero.

Fy(x,y,λ)=y[x2+y2λ(2x4y+5)]0=2y+4λy=2λ

The partial derivatives of F with respect to λ and then set it to zero

### 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

#### In Exercises 516, evaluate the given quantity. log5125

Finite Mathematics and Applied Calculus (MindTap Course List)

#### 165/8161/2167/8

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

#### In Problems 7-34, perform the indicated operations and simplify. 29.

Mathematical Applications for the Management, Life, and Social Sciences

#### Given: 13 Prove: mn

Elementary Geometry for College Students

#### True or False: , for –π ≤ t ≤ π, is smooth.

Study Guide for Stewart's Multivariable Calculus, 8th

#### Graph each equation. x=2

College Algebra (MindTap Course List) 