I’m trying to prove the Arithmetic-Geometric-Mean Inequality (A-G-M) using Lagrange multipliers. For positive real numbers $ x_{1},x_{2},\ldots,x_{n} $, we want to show that
$$
(x_{1} x_{2} \cdots x_{n})^{1/n} \leq \frac{x_{1} + x_{2} + \cdots + x_{n}}{n}.
$$
Consider the function $ f(x_{1},x_{2},\ldots,x_{n}) = x_{1} + x_{2} + \cdots + x_{n} $ subject to the constraint
$$
x_{1} x_{2} \cdots x_{n} = c,
$$
where $ c $ is a constant.
So I’m using Lagrange multipliers to solve this. I get $ \dfrac{1}{n} = \dfrac{\lambda c}{x_{i}} $ for all $ 1 \leq i \leq n $. Then $ \dfrac{x_{i}}{n} = \lambda c $. Then the sum of all the $ \dfrac{x_{i}}{n} $-terms yields $ x_{i} = \lambda cn $. I’m not sure where to go from here, or if I’ve made a mistake somewhere. Any tips?
Best Answer
Your result shows that all the $x_i$ are equal to $n \lambda c$, so their product is $(n \lambda c)^n$.
If this equals $c$, $c = (n \lambda c)^n$ so $\lambda = c^{-1+1/n}/n$ and the sum of the $x_i$ is $n^2 \lambda c =n^2 c (c^{-1+1/n}/n) =n c^{1/n} $ and each $x_i$ is $c^{1/n} $.