For $x_i\ge0$, where $i=1,2,...,n$, satisfying
$\sum_{i=1}^n\,x_i^2+2\,\sum_{1\le k
find the maximum and minimum of $\sum_{i=1}^n x_i.$
For $x_i\ge0$, where $i=1,2,...,n$, satisfying
$\sum_{i=1}^n\,x_i^2+2\,\sum_{1\le k
find the maximum and minimum of $\sum_{i=1}^n x_i.$
Actually the minimum is pretty trivial:
$\left(\sum_{i=1}^n\,x_i\right)^2\ge \sum_{i=1}^{n}\,x_{i}^{2}+2\,\sum_{1\le k
The maximum is much more interesting.
Let $y_i:=\frac{x_i}{\sqrt{i}}$ for every $i=1,2,\ldots,n$. Then, $1=\sum_{i=1}^n\,x_i^2+2\,\sum_{1\leq k
(1) $y_i=\frac{2\sqrt{i}-\sqrt{i+1}-\sqrt{i-1}}{\lambda}$, or $x_i=\frac{2i-\sqrt{i(i+1)}-\sqrt{i(i-1)}}{\lambda}$ for all $i=1,2,\ldots,n-1$, and
(2) $y_n=\frac{\sqrt{n}-\sqrt{n-1}}{\lambda}$, or equivalently, $x_n=\frac{n-\sqrt{n(n-1)}}{\lambda}$.