Minimum value of algebraic expression. 
If $0\leq x_{i}\leq 1\;\forall i\in \left\{1,2,3,4,5,6,7,8,9,10\right\},$ and $\displaystyle \sum^{10}_{i=1} x^2_{i}=9$
Then $\max$ and $\min$ value of $\displaystyle \sum^{10}_{i=1} x_{i}$

$\bf{My\; Try::}$
Using Cauchy-Schwarz Inequality
$$\left(x^2_{1}+x^2_{2}+.......+x^2_{10}\right)\cdot (1^2+1^2+....1^2)\geq \left(x_{1}+x_{2}+....+x_{10}\right)^2$$
So we get $$\left(x_{1}+x_{2}+....+x_{10}\right)\leq \sqrt{90}$$
Now How can I calculate for its minimum value, Help required, Thanks

You have already got the right inequality for the maximum, all you need to add is that equality can be achieved when $x_i = 3/\sqrt{10}$. 
For the minimum, note $x_i\in [0,1]\implies x_i^2\leqslant x_i\implies 9=\sum x_i^2\leqslant \sum x_i$  Equality is possible here when one of the $x_i$ is $0$ and all others $1$. 
