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Chapter 1: Q10-14P (page 1)

Do Example 3b using Lagrange multipliers.

Short Answer

Expert verified

Hence, the points on the given surface nearest to the origin are 0,0,1and0,0,-1.

Step by step solution

01

Lagrange's Multipliers

Lagrange's multipliers is a method of optimization to find the local maxima and minima of the function fx,y,z with constraint φ, which is given by:

Fx,y,z=f+λφ

02

Find derivatives

The given functions are:

fx,y,z=x2+y2+z2φx,y,z=z2-x2=1

Now, using Lagrange's multipliers method, we get:

Fx,y,z=f+λφ=x2+y2+z2+λz2-x2

To find the minimum or maximum angle, we have:

∂F∂x=2x-2xλ=0∂F∂y=2y=0∂F∂z=2z+2zλ=0

03

Find the values of variables

From ∂F∂x=0, we get:

x=0orλ=1⇒z=±1orz=0,x=±i

From ∂F∂y=0, we get:

y=0

From ∂F∂z=0, we get:

z=0orλ=-1⇒x=±iorx=0,z=±1

Taking the real values of points, the point of interest obtained is role="math" localid="1664881443001" 0,0,±1.

Hence, the points on the given surface nearest to the origin are 0,0,1and0,0,-1.

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