# 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?

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

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

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

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

always 1 ... so there is no max or min ? its just flat?
f = ∑ xi. Why is it 1??

ehild

...i don't know ...

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

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