Maximum value of a product How to write the number $60$ as $\displaystyle\sum^{6}_{i=1} x_i$ such that $\displaystyle\prod^{6}_{i=1} x_i$ has maximum value?

Thanks to everyone :)
Is there a way to solve this using Lagrange multipliers?

You can prove it without explicitly invoking the AM-GM inequality by first supposing that the product is not maximal when $x_i = 10\ \forall i$. Then, when the product is maximal, $x_i \neq x_j$ for some $i,j$. Then one of $x_i$ and $x_j$ is larger; the other smaller; suppose $x_i < x_j$. The proof is completed by showing that letting $x_i \rightarrow x_i+1$ and $x_j \rightarrow x_j-1$ increases the product, which contradicts the assumption that the product was maximal.
