   Chapter 14.8, Problem 45E

Chapter
Section
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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Evaluate the integral by making the given substitution. x2x3+1dx,u=x3+1

Single Variable Calculus: Early Transcendentals, Volume I

Convert the expressions in Exercises 8596 radical form. x1/3y3/2

Finite Mathematics and Applied Calculus (MindTap Course List)

Evaluate the integral. 19xdx

Calculus (MindTap Course List)

The difference between population variance and sample variance.

Statistics for The Behavioral Sciences (MindTap Course List)

In Exercises 914, evaluate the expression. 12. (8)5/3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Fill in each blank: 6gal3qt=pat

Elementary Technical Mathematics

The equation that best describes the curve at the right is:

Study Guide for Stewart's Multivariable Calculus, 8th

Sometime, Always, or Never:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 