Extreme values of function

Question

We have $A = \overline B(O_3, 1)$\ {$O_3$} and $f:A \to \mathbb R$, $f(x,y,z) = \frac{x+y+z}{\sqrt{x^2+y^2+z^2}}$. The problem asks for extreme values of function $f$ and asks if $f$ is touches them.

Answer 1

The function is infinitely many times continuous differentiable over $\Bbb R-\{0\}$, therefore at any extreme point we have$$\nabla f=0\iff \left({\partial f\over \partial x},{\partial f\over \partial y},{\partial f\over \partial z}\right)=0 $$ therefore$$x^2+y^2+z^2=x(x+y+z)\\x^2+y^2+z^2=y(x+y+z)\\x^2+y^2+z^2=z(x+y+z)$$for $x+y+z=0$ we have $x=y=z=0$ and for $x+y+z\ne 0$ by dividing both sides on $x+y+z$ we obtain$$x=y=z$$then the extreme points over $A$ are on $\{(x,x,x) \ \ |\ \ 0<x<1\}$ for which $f$ touches them.