$x,y,z$ are all strictly positive, $x+y+z=1$, what is $\max(xyz)$

Question

$x,y,z$ are all strictly positive, $x+y+z=1$, what is $\max(xyz)$?

My attempt:

Using rand() function in Microsoft Excel to generate random numbers between $0$ and $1$. I used this function for the values of $x$ and $y$.

For the value of $z$, I used the formula $z=1-x-y$. This will make some values to be negative, which does not satisfy the condition given in the problem statement. However, repeating the process will lead us to find positive $z$ values.

Then I used max() function. I observed that the $\max(xyz)=0.03703…$

I am not sure if $0.03703…$ is really the maximum value of the product of $x,y,$ and $z$.

How to find the exact value (closed form) of $\max(xyz)$ without using programs?

Any help will be appreciated. Thanks!

Answer 1

Hint: For $x,y,z>0$,$$xyz\leq\left(\frac{x+y+z}{3}\right)^3.$$ When does the equality hold?