<s>[INST] Minimize $P=5\left(x^2+y^2\right)+2z^2$ For $\left(x+y\right)\left(x+z\right)\left(y+z\right)=144$, minimize $$P=5\left(x^2+y^2\right)+2z^2$$
I have no idea. Can you make a few suggestions? 
 [/INST] By AM-GM 
$1728=(3x+3y)(2x+2z)(2y+2z)\leq \left(\frac{5x+5y+4z}{3}\right)^3$
$\Rightarrow 5x+5y+4z\geq 36$
By Cauchy-Schwarz we have:
$18P=(5x^2+5y^2+2z^2)(5+5+8)\geq (5x+5y+4z)^2\geq 36^2$
$$\Rightarrow P\geq 72$$
Hence $P_{\min}=72\Leftrightarrow (x,y,z)=(2,2,4)$
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