Suppose you have to find the maximum value of the product $xyz$, with the following conditions:
- $x,y,z \in \mathbb{R}$ and all positive
- $x+y+z=a\,, \quad a\gt0$
Intuitively, the solution should be $x=y=z=a/3$, but I was wondering if it was possible to solve this problem without resorting to multivariable calculus, e.g. by algebraic manipulation of the expression $xy(a-x-y)$, in a similar way as with the univariate quadratic case $x^2+bx+c$. I'm particularly interested in the general method and line of reasoning for solving this kind of problems; any help would be appreciated!