Question

For $x$, $y$, and $z$ positive real numbers, what is the maximum possible value for


\[\sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}}+ \sqrt{\frac{2z+3x}{6x+5y+4z}}\]

Answer 1

The site uses some fancy MathJax plugin or something :)

No $'s I guess.
Use `\[` to start math mode and `\]` to end.
If you use the equation button at the bottom,
it will automatically include these two end brackets for you.
Example:\[\sqrt{\frac{3x+4y}{6x+5y+4z}}\]

If you ever need to do in-line math code,
instead use `\(` and `\)`.

Example \(\large\rm x^2+6\) given here.

Answer 2

\(\large\rm \color{royalblue}{\text{Welcome to OpenStudy! :)}}\)

Answer 3

A local maximum by brute force is
x=0.2856444350572444
y=0.8569333051717334
z=1.7138666103434666
and the maximum value is \(\sqrt 3\), to about 34 significant digits.

Answer 4

could you elaborate a bit? @mathmate

Answer 5

Well, I had calculated the partial derivatives, which guided my program to make little steps in the right directions. I started with (1,1,1) and ended with the above values, and the value of the expression corresponded to sqrt(3) to about 34 figures as I refined the values of x, y and z.
As I said, it is a local maximum, and not a global maximum.
I was hoping someone else would be working on it to give an analytical answer.
I tried starting point as (100,100,100) and I still got sqrt(3) as the maximum value.