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 .

Observe that we can formulate this question in terms of linear algebra.  Note the condition that $a^2 + b^2 + c^2 = 48$ is the same as a vector with norm $\sqrt{48} = 4\sqrt{3}$.  The condition of maximizing $a+b+c$ is the same as maximizing the inner product of $\mathbb{1} = \left [ \begin{array}{ccc}
1&1&1\\
\end{array}\right ]$ with the vector $v =\left [ \begin{array}{ccc}
a&b&c\\
\end{array} \right ]$.  We know however by Cauchy-Schwarz that $\langle \mathbb{1}, v\rangle$ is maximized when $v$ is linearly dependent with $\mathbb{1}$. In other words we would like $v$ to be a multiple of $\mathbb{1}$ hence $a=b=c$, such that $3a^2 = 48$, or rather $a = 4$.
