Lagrange Multipliers(just need confirmation)

  • Thread starter arl146
  • Start date
  • #1
343
1

Homework Statement


Use Lagrange multipliers to find the max and min values of the function subject to the given constraints:

f(x1,x2,...,xn) = x1 + x2 + ... + xn
constraint: (x1)^2 + (x2)^2 + ... (xn)^2 = 1


The Attempt at a Solution



fo x1 to xn values, x must equal 1/sqrt(n) in order to equal 1. [ g(x)=k --> the constraint ]

so (1/sqrt(n))^2 + (1/sqrt(n))^2 + (1/sqrt(n))^2 + ETC = 1

right? so how else could i answer that? how would a give a value for the min/max?
 

Answers and Replies

  • #2
ehild
Homework Helper
15,543
1,909
fo x1 to xn values, x must equal 1/sqrt(n) in order to equal 1. [ g(x)=k --> the constraint ]
The sum of xi2 has to be equal to 1. So you have to solutions xi=1/√n or xi=-1/√n.

ehild
 
  • #3
343
1
but what about the max/min .... or did i technically answer that?
 
  • #4
ehild
Homework Helper
15,543
1,909
but what about the max/min .... or did i technically answer that?
What is the value of the function f if all xi-s are 1/√n ? and when all xi=-1/√n ?

ehild
 
  • #5
343
1
always 1 ... so there is no max or min ? its just flat?
 
  • #6
ehild
Homework Helper
15,543
1,909
always 1 ... so there is no max or min ? its just flat?
f = ∑ xi. Why is it 1???????

ehild
 
  • #7
343
1
...i dont know ....
 
  • #8
ehild
Homework Helper
15,543
1,909
Say n=2. f(x1,x2)=x1+x2. The constraint is x12+x22=1.
You found that the extrema are when x1=x2=1/√2 or x1=x2=-1/√2. Substitute back to f:

f=x1+x2=1/√2+1/√2=2/√2 or

f=x1+x2=-1/√2+(-1/√2)=-2/√2

Are the values of f equal to 1?

ehild
 
  • #9
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
always 1 ... so there is no max or min ? its just flat?
Is the function x1 + x2 + ...+ xn constant for all (x1, x2, ...,xn) on the sphere? Look at the cases of n = 2 and n = 3.

RGV
 

Related Threads on Lagrange Multipliers(just need confirmation)

Replies
14
Views
734
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
976
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
1K
Top