<s>[INST] Minimum Value of expression Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of:
$$x^2+4xy+4y^2+2z^2$$
Perhaps AM-GM and manipulation but I'm not quite sure how?
Source BMO.
 [/INST] Yes AM-GM is the right approach.
By AM-GM,
$$\frac{x^2+2xy+2xy+4y^2+z^2+z^2}{6} \geq (16 \cdot x^4 \cdot y^4 \cdot z^4)^{\frac{1}{6}}=$$
$$=(2^{24})^{\frac{1}{6}}=16$$ with equality iff $x=z=2y$.
So the minimum value of $x^2+4xy+4y^2+2z^2$ is $16 \cdot 6=96$. QED
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