[Math] Find the maximum value of the product xyz

Question

IF $x , y , z$ are arbitary positive real numbers satisfying the equation

$$ 4xy + 6yz + 8xz = 9$$

Find the maximum value of the product $xyz$.

I dont know from where to begin .

3 variables and one equation.

How I can achieve this?

Answer 1

Arithmetic mean is $\ge$ Geometric mean, i.e. $${{4xy + 6yz + 8xz}\over3} \ge {{(4xy\cdot 6yz\cdot 8xz)}}^{1/3}.$$