Find the minimum value of the product $P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when $xyz=1$ and $x,y,z$ are positive real numbers.
I don't know how to go about this. AM-GM got really messy, and I don't know how to incorporate harmonic and quadratic root means.
Best Answer
By AM-GM we have $$(2x+3y)(x+3z)(y+2z)\geq 8\sqrt{36(xyz)^2}=48$$ since $$\frac{2x+3y}{2}\geq \sqrt{6xy}$$ etc