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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.

31. Exercise 41

41. Find the shortest distance from the point (2, 0, -3) to die plane x + y + z = 1.

To determine

To find: The shortest distance from the point (2,0,3) to the plane x+y+z=1 by using Lagrange multipliers.

Explanation

Given:

The plane x+y+z=1.

Definition 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 shortest distance between the point (2,0,3) and (x,y,z).

d=(x2)2+(y0)2+(z+3)2d2=(x2)2+y2+(z+3)2

Thus, the minimum value of the function d2=f(x,y,z)=(x2)2+y2+(z+3)2 subject to the constraint g(x,y,z)=x+y+z=1.

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

f(x,y,z)=λg(x,y,z)fx,fy,fz=λgx,gy,gzfx[(x2)2+y2+(z+3)2],fy[(x2)2+y2+(z+3)2],fz[(x2)2+y2+(z+3)2]=λgx(x+y+z),gy(x+y+z),gz(x+y+z)2(x2),2y,2(z+3)=λ1,1,1

Thus, the value of f(x,y,z)=λg(x,y,z) is 2(x2),2y,2(z+3)=λ1,1,1

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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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