Maximizing $f$ in $\mathbb{R}^3$ 
Find the domain and the maximum value that the function
  $$f(x,y,z)=\frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}}$$
  may attain in its domain.

I have found the domain of the function to be $\mathbb{R^3\backslash\mathbf{0}}$. To maximize I differentiated in terms of $x,y,z$ having
$$f_x=\frac{-2 x y-3 x z+y^2+z^2}{\left(x^2+y^2+z^2\right)^{3/2}},\quad f_y=\frac{2 x^2-x y+z (2 z-3 y)}{\left(x^2+y^2+z^2\right)^{3/2}},\quad f_z=\frac{3 \left(x^2+y^2\right)-z (x+2 y)}{\left(x^2+y^2+z^2\right)^{3/2}}$$
But to solve the system $f_x=0,f_y=0$ and $f_z=0$ is rather hard. What are the plausible values of $x,y,z$?

Consider the line
$x = t, y = 2t, z = 3t$
Along this line.
$f(t,2t,3t) = \frac {14 t}{\sqrt{14 t^2}} = \sqrt {14}$
let's find some orthogonal vectors.
$\mathbf u = (1,2,3)\\
\mathbf v = (2,-1,0)\\
\mathbf w = (0,3,-2)$
Any point in $\mathbb R^3$ is some linear combination
$c_1\mathbf u + c_2\mathbf v+ c_3\mathbf w$
$f(c_1\mathbf u + c_2\mathbf v+ c_3\mathbf w) = \frac {14c_1}{\sqrt{14c_1^2 + 5c_2^2 + 13c_3^2}}\\
|f|\le \sqrt 14 \text{ sgn}(c_1)$
