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

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Chapter 7.6, Problem 10E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 1-12, use Lagrange multipliers to find the indicated extremum. Assume that x and y are positive. See Example 1.

Minimize f ( x , y ) = x 2 + y 2

Constraint: 2 x + 4 y − 15 = 0

To determine

To calculate: The minimum value of the function f(x,y)=x2+y2 subject to the constraint 2x+4y−15=0 by the use of Lagrange multipliers.

Explanation

Given Information:

The provided function is f(x,y)=x2+y2 subject to the constraint 2x+4y−15=0. 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)=0 Fλ(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 2x+4y−15=0.

So, g(x,y)=2x+4y−15

Now, the new function F is,

F(x,y,λ)=x2+y2−λ(2x+4y−15)

Differentiate above equation with respect to x, y and λ,

Fx(x,y,λ)=2x2x2+y2−2λFy(x,y,λ)=2y2x2+y2−4λ

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Prove the statement using the , definition of a limit and illustrate with a diagram like Figure 9. 18. limx2(3...

Single Variable Calculus

Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulb...

Probability and Statistics for Engineering and the Sciences

For Problems 5-54, perform the following operations with real numbers. Objectives 3-6 (5.4)(7.2)

Intermediate Algebra

Simplify each expression in Exercises 1730, expressing your answer in positive exponent form. (x2y1z0xyz)2

Finite Mathematics

Difference of Sets Another operation that can be defined on sets A and B is the difference of the sets, denoted...

Mathematical Excursions (MindTap Course List)

In problems 15-22, simplify by combining like terms. 19.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate the integral. 33. xsecxtanxdx

Single Variable Calculus: Early Transcendentals

Write each of the following in terms of sin and cos ; then simplify if possible. csccotcos

Trigonometry (MindTap Course List)

Write an inequality for each graph:

Elementary Technical Mathematics

Catalina Jewelers has 147.25 ounces of 14-carat gold in stock. a. How many custom necklaces can be manufactured...

Contemporary Mathematics for Business & Consumers

5. Prove that multiplication is associative in

Elements Of Modern Algebra

In Exercises 29 to 31, M and Nare the midpoints of sides RS and RT ofRST, respectively. Given:MN=2y-3 ST=3y Fin...

Elementary Geometry For College Students, 7e

Area In Exercises 31 and 32, sketch a graph of the region bounded by the graphs of the equations. Then use a do...

Calculus: Early Transcendental Functions (MindTap Course List)

Determining Concavity In Exercises 5-16, determine the open intervals on which the graph of the function is con...

Calculus of a Single Variable

Draw the graph of 4x-3y=9 by first changing the equation to Slope-Intercept Form.

Elementary Geometry for College Students

In Exercise 1-6, express each equation in logarithmic form. 26=64

Finite Mathematics for the Managerial, Life, and Social Sciences

True or False: If f is twice differentiable and x0 is an inflection point for f, then the curvature of f at x0 ...

Study Guide for Stewart's Multivariable Calculus, 8th

Minimum Length A legal-sized sheet of paper (8.5 inches by 14 inches) is folded so that comer P touches the opp...

Calculus (MindTap Course List)

Solving an Exact Differential Equation In Exercises 714, verify that the differential equation is exact. Then f...

Multivariable Calculus

For f(x) = 3x + 1 and g(x) = x, ?

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

Circles Two circles of radius 4 are tangent to the graph of y2=4x at the point (1,2). Find equations of these t...

Calculus

Describe quota sampling, recognize examples of this technique in research reports, and explain why it is used.

Research Methods for the Behavioral Sciences (MindTap Course List)

Applications Use scientific notation to compute each answer. Write all answer in scientific notation. New way t...

College Algebra (MindTap Course List)

Identify the advantages of a multiple-group design compared to an experiment with only two groups.

Research Methods for the Behavioral Sciences (MindTap Course List)

Determine the sine, cosine, or tangent functions of the following angels. Round the answers to 5 decimal places...

Mathematics For Machine Technology

How many great-great grandparents does a drone bee have ? How is each related to the drone.

Mathematics: A Practical Odyssey

Dividend yield is the annual dividend paid by a company expressed as a percentage of the price of the stock (Di...

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

In 2010, the National Football League adopted new rules designed to limit head injuries. In a survey conducted ...

Introduction To Statistics And Data Analysis

The following payoff table shows profit for a decision analysis problem with two decision alternatives and thre...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

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