$x,y,z$ 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?

Answer 2

Another hint, aiming in the same direction as the answer from Feng Shao:

If you take $x,y,z$ to be the edges of a cuboid in 3d space, how can you maximize the volume of this guy when the sum of edge lengths are fixed? Can you prove that a cube (all sides have equal length) has the best ratio of volume to sum of side lengths?

Answer 3

There are three ways that I can come up with.

1. By using the AM-GM inequality, you have

$$\sqrt [3]{xyz} \leq \frac{x+y+z}{3},$$

for non-negative $x$, $y$, and $z$.

2. 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.

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