   Chapter 14.8, Problem 33E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.33. Exercise 4343. 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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