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

There are three ways that I can come up with.


*

*By using the AM-GM inequality, you have


$$\sqrt [3]{xyz} \leq \frac{x+y+z}{3},$$
for non-negative $x$, $y$, and $z$.


*As you attempted, represent $z$ with $x$ and $y$, and compute partial derivatives of $xyz$ with $z$ being replaced by $1-x-y$, and check if the critical points are in the domain and the signs of second partial derivatives.

*The method of Lagrange multipliers:
https://en.m.wikipedia.org/wiki/Lagrange_multiplier
