maximum value of $a+b+c$ given $a^2+b^2+c^2=48$? How can i get maximum value of this  $a+b+c$ given $a^2+b^2+c^2=48$ by not  using AM,GM and lagrange multipliers .

You can use the identity
$$(a+b+c)^2=3(a^2+b^2+c^2)-(a-b)^2-(b-c)^2-(c-a)^2$$
The right-hand side is clearly maximised when $a=b=c$ and then $a+b+c=12$
