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 .

By the arithmetic-quadratic mean
inequality,
$\frac{a+b+c}{3}
\le \sqrt{\frac{a^2+b^2+c^2}{3}}
= \sqrt{\frac{48}{3}}
=\sqrt{16}
=4
$
with equality iff
$a=b=c$.
Therefore
$a+b+c = 12$
and
$a=b=c = 4$.
