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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Consider the problem of maximizing the function f(x, y) = 2x + 3y subject to the constraint x + y = 5.

(a) Try using Lagrange multipliers to solve the problem.

(b) Does f(25, 0) give a larger value than the one in part (a)?

(c) Solve the problem by graphing the constraint equation and several level curves of f.

(d) Explain why the method of Lagrange multipliers fails to solve the problem.

(e) What is the significance of f(9, 4)?

(a)

To determine

To find: The extreme value of the function f(x,y)=2x+3y subject to the constraint x+y=5 by using Lagrange multipliers.

Explanation

Given:

The function is f(x,y)=2x+3y subject to the constraint x+y=5 .

Result used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx , fy=λgy , fz=λgz and g(x,y,z)=k ”.

Calculation:

The Lagrange multipliers f(x,y)=λg(x,y) is computed as follows,

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(2x+3y),fy(2x+3y)=λgx(x+y5),gy(x+y5)2,3=λ12x,12y

Thus, the value of f(x,y)=λg(x,y) is 2,3=λ12x,12y

(b)

To determine

To compare: The values of f(25,0) , the value of f(9,4) from part (a) and find which is greater.

(c)

To determine

To sketch: The constraint equation x+y=5 and the several level curves of the function f.

(d)

To determine

To explain: The failure of Lagrange multipliers for the given problem.

(e)

To determine

To find: The significance of f(9,4) .

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Chapter 14 Solutions

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