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