Hi
Suppose that $n$ is a certain Natural number and $\{b_i\}_{i=0,..,n-1}$ is a certain array of positive real numbers, find array $\{a_i\}_{i=0,..,n-1}$ of positive real numbers such that $\sum_{i=0}^{n-1} a_i = 1$ and $\sum_{i=0}^{n-1} \sqrt {a_i b_i}$ is the maxmum possible value.
I think thats maximum when all $\frac{a_i}{b_i}$ ratios are equal. But I need to make sure.
find array that this sum is the maximum
4
$\begingroup$
algebra-precalculus
-
0You may not belive that this problem came out from working on a software design (a model for internet sever provider and bandwith management). I am now a programmer and I was many years far from pure mathematics, and forgot most of these iniqality techniques (I remind how much important and applicable was cauchy-schwarz) – 2011-04-10
2 Answers
3
You are right: by Cauchy-Schwarz inequality, $\sum\sqrt{a_ib_i}\leq\sqrt{\sum a_i}\sqrt{\sum b_i}$, with equality iff $a_i=\lambda b_i$ for some positive $\lambda$.
edit: oops, I thought you were looking for the maximum. Minimum is when all $a_i$'s are $0$, except for the $i$ where $b_i$ is minimal (so there $a_i=1$).
-
0yeah. maximum. I was going to say oops!! but saw your answer. you are right and I got my answer (I edit post and change to maximum). thanks. – 2011-04-09
1
Your conjecture is correct (key term: Cauchy-Schwarz Inequality).