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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The plane x + y + 2z = 2 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

To determine

To find: The nearest and farthest point from the origin of an ellipse

Explanation

Given:

The plane x+y+2z=2 intersects the paraboloid z=x2+y2 in an ellipse.

Definition used:

“The Lagrange multipliers defined as f(x0,y0,z0)=λg(x0,y0,z0)+μh(x0,y0,z0) . This equation can be expressed as fx=λgx+μhx , fy=λgy+μhy , fz=λgz+μhz and g(x,y,z)=k , h(x,y,z)=c ”.

Calculation:

Let the equation of an ellipse be f(x,y,z)=x2+y2+z2 .

Thus, the maximize function f(x,y,z)=x2+y2+z2 is subject to the constraints g(x,y,z)=x+y+2z=2 and h(x,y,z)=x2+y2z=0 .

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

f(x,y,z)=λg(x,y,z)+μh(x,y,z)fx,fy,fz=λgx,gy,gz+μhx,hy,hzfx(x2+y2+z2),fy(x2+y2+z2),fz(x2+y2+z2)=λgx(x+y+2z),gy(x+y+2z),gz(x+y+2z)+μhx(x2+y2z),hy(x2+y2z),hz(x2+y2z)2x,2y,2z=λ1,1,2+μ2x,2y,1

Thus, the value of f(x,y,z)=λg(x,y,z)+μh(x,y,z) is 2x,2y,2z=λ1,1,2+μ2x,2y,1 .

The result, 2x,2y,2z=λ1,1,2+μ2x,2y,1 can be expressed as,

2x=λ+2μx (1)

2y=λ+2μy (2)

2z=2λμ (3)

Subtract the equation (2) from (1),

2x2y=(λ+2μx)(λ+2μy)2(xy)=2μ(xy)

Since, 2(xy)=2μ(xy) , there are two possibilities such as either x=yorμ=1

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Chapter 14 Solutions

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