1. ## Lagrange Multipliers

Use Lagrange multipliers to find three positive numbers whose sum is 15 and whose product is as large as possible.

2. Originally Posted by CandyKanro
Use Lagrange multipliers to find three positive numbers whose sum is 15 and whose product is as large as possible.

Let $\displaystyle x_1, x_2 , x_3$ be the numbers:

$\displaystyle f = x_1 x_2 x_3 - \lambda(x_1 + x_2 +x_3 - 15)$

So you will get three conditions from this, $\displaystyle x_2 x_3 = x_1 x_3 = x_1 x_2 = \lambda$. This yields $\displaystyle x_1 = x_2 = x_3$ using the fact thay are positive. Substituting in the constaint yields $\displaystyle x_1 = x_2 =x_3 = 5$. Thus the maximum value is of product is $\displaystyle 5^3$

(Note that I did not add the constraint that they are positive, and the answer turned out to be positive... so no problem with that)

3. Originally Posted by Isomorphism
So you will get three conditions from this, $\displaystyle x_2 x_3 = x_1 x_3 = x_1 x_2 = \lambda$. This yields $\displaystyle x_1 = x_2 = x_3$ using the fact thay are positive. Substituting in the constaint yields $\displaystyle x_1 = x_2 =x_3 = 5$. Thus the maximum value is of product is $\displaystyle 5^3$

(Note that I did not add the constraint that they are positive, and the answer turned out to be positive... so no problem with that)

i dont get this part =S

can u specify it further? =(