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!

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    $\begingroup$ Lookup the AM-GM inequality. $\endgroup$ – dxiv Jul 9 '18 at 23:38
  • $\begingroup$ @dxiv I'm sorry, I don't get it. Could you tell me where to look in particular? $\endgroup$ – The Footprint Jul 9 '18 at 23:47
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    $\begingroup$ The link takes you to $\,\sqrt[3]{xyz} \le \dfrac{x+y+z}{3}\,$ with equality iff $x=y=z$, which solves the case you posted. $\endgroup$ – dxiv Jul 9 '18 at 23:50
  • $\begingroup$ Thank you, that's great. How does one see it at a glance? $\endgroup$ – The Footprint Jul 10 '18 at 0:09
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    $\begingroup$ Practice, there really is no substitute or shortcut. Once you've seen a few, you'll recognize them more quickly. $\endgroup$ – dxiv Jul 10 '18 at 0:15

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