Let x, y, and z be real numbers such that \(x + y + z = 0\) and \(xyz = 2\).

Find the maximum value of \(x^3 y + y^3 z + z^3 x\).

I honestly do not know where to start this problem because of this reasoning here:

Basically, we can deduce that one of the variables must be positive, and the other two negative, and because of this, the QM-AM-GM-HM inequalities have reduced functionality.

Perhaps the Cauchy-Schwarz can be of use here?

Thanks for any help!

EnchantedLava68 Jul 3, 2021