$$\frac{x}{\sqrt{x+y}} + \frac{y}{\sqrt{y+z}} + \frac{z}{\sqrt{z+x}}$$ Over all non-negative $x,y,z$ satisfying $x+y+z=4$, let the maximum value of the above expression be $M$. What is the value of $\lfloor 10000M \rfloor$?
This is a question from here, and I decided to give it a go, bit got stuck. Here is my approach: Let $x+y=u^2, y+z=v^2, z+x=w^2$. Hence, $u^2 + v^2 + w^2 =8$. Also, $x=4-v^2, y=4-w^2, z=4-u^2$ and the expression becomes $$\frac{4}{u} + \frac{4}{v} + \frac{4}{w} - \frac{v^2}{u} - \frac{w^2}{v} - \frac{u^2}{w} $$ I was itching to use Titu's Lemma/Cauchy-Schwarz in Engel form but then I realized that it will help provide a lower bound. With that ruled out, I am thinking of Rearrangement Inequality but my mind is pulling off a perfect blank. Any guidance will be welcome.
Note: The answer to the question is $25000$.