Lagrange Multipliers(just need confirmation)

[arl146]

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?

 

[ehild]

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

The sum of x_i² has to be equal to 1. So you have to solutions x_i=1/√n or x_i=-1/√n.

ehild

 

[arl146]

but what about the max/min ... or did i technically answer that?

 

[ehild]

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

 

[arl146]

always 1 ... so there is no max or min ? its just flat?

 

[ehild]

always 1 ... so there is no max or min ? its just flat?

f = ∑ x_i. Why is it 1??

ehild

 

[arl146]

...i don't know ...

 

[ehild]

Say n=2. f(x₁,x₂)=x₁+x₂. The constraint is x_1²+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

 

[Ray Vickson]

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