   Chapter 14, Problem 19RCC

Chapter
Section
Textbook Problem

Explain how the method of Lagrange multipliers works in finding the extreme values of f(x, y, z) subject to the constraint g(x, y, z) = k. What if there is a second constraint h(x, y, z) = c?

To determine

To explain: The method of Lagrange multipliers for finding the extreme values f(x,y,z) subject to the constraint g(x,y,z)=k ; what happens if there is a second constraint h(x,y,z)=c .

Explanation

Step 1:

Consider the function f(x,y,z) .

Find the gradient vector of the function f(x,y,z) .

f(x,y,z)=fxi+fxj+fxk=fx,fy,fz

Find the gradient vector of the function g(x,y,z)=k .

g(x,y,z)=gxi+gxj+gxk=gx,gy,gz

Step 2:

Comparing the following equations as follows,

f(x,y,z)=λg(x,y,z)

fx=λgx (1)

fy=λgy (2)

fz=λgz (3)

g(x,y,z)=k (4)

Step 3:

From the equations (1),(2),(3) and (4),

Compute the values of x,yandz .

Substitute the values of x,yandz in the equation f(x,y,z) and obtain the absolute maximum and absolute minimum values.

This is the required solution.

If there are two constraints follow the below method.

Step 1:

Consider the function f(x,y,z) .

Find the gradient vector of the function f(x,y,z) .

f(x,y,z)=fxi+fxj+fxk=fx,fy,fz

Find the gradient vector of the function g(x,y,z)=k

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