Answer
$x=y=z=\frac{5 \sqrt{3}}{3}$
Work Step by Step
We know:
\[
\sqrt{x^{2}+y^{2}+z^{2}}=5 \equiv x^{2}+y^{2}+z^{2}=25
\]
Let $f(x, y, z)=x+y+z$ be the sum of the vector components. We need to maximize $f$ under the constraint $x^{2}+y^{2}+z^{2}=25 .$ Using Lagrange Multipliers:
\[
\begin{aligned}
\nabla f=& \lambda \nabla g \Rightarrow(1,1,1)=\lambda\langle 2 x, 2 y, 2 z\rangle \\
\lambda &=\frac{1}{2 x}=\frac{1}{2 y}=\frac{1}{2 z} \Rightarrow x=y=z
\end{aligned}
\]
Using this in the given constraint:
\[
x^{2}+y^{2}+z^{2}=25 \Rightarrow 3 x^{2}=25 \Rightarrow x=y=z=\pm \frac{5 \sqrt{3}}{3}
\]
We note that for $x=y=z=\frac{5 \sqrt{3}}{3},$ we obtain the largest component sum.
