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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.

33. Exercise 43

43. Find the point on the cone z2 = x2 + y2 that are closest to the point (4, 2, 0).

To determine

To find: The closest point from the point (4,2,0) to the cone z2=x2+y2 by using Lagrange multipliers.

Explanation

Given:

The equation of the cone is z2=x2+y2.

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 (4,2,0) and (x,y,z).

d=(x4)2+(y2)2+(z0)2d2=(x4)2+(y2)2+z2

Thus, the minimize function d2=f(x,y,z)=(x4)2+(y2)2+z2 subject to the constraint g(x,y,z)=x2+y2z2=0.

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[(x4)2+(y2)2+z2],fy[(x4)2+(y2)2+z2],fz[(x4)2+(y2)2+z2]=λgx(x2+y2z2),gy(x2+y2z2),gz(x2+y2z2)2(x4),2(y2),2z=λ2x,2y,2z

Thus, the value of f(x,y,z)=λg(x,y,z) is 2(x4),2(y2),2z=λ2x,2y,2z

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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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