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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 minimizing the function f(x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform).

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

(b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition ∇f(0, 0) = λ∇g (0, 0) is not satisfied for any value of λ.

(c) Explain why Lagrange multipliers fail to find the minimum value in this case.

(a)

To determine

To find: The extreme value of the function f(x,y)=x subject to the constraint y2+x4x3=0 by using Lagrange multipliers.

Explanation

Given:

The function is f(x,y)=x subject to the constraint y2+x4x3=0 .

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 given function is, f(x,y)=x and g(x,y)=y2+x4x3 .

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

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(x),fy(x)=λgx(y2+x4x3),gy(y2+x4x3)1,0=λ4x

(b)

To determine

To show: The minimum value of f(0,0)=0 and find the Lagrange condition f(0,0)=λg(0,0) does not exist.

(c)

To determine

To explain: The failure of the Lagrange multipliers while finding the minimum values.

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Chapter 14 Solutions

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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

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