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Minimum Distance Optimization

Let $ x, $ $ y, $ and $ z $ be positive real numbers such that $ xyz = 2 $. Find the minimum value of \[x^4 + 4y^2 + 4z^4.\]

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$\frac{a}{b}$
.
0
=
%

$a^n$
$a^{\circ}$
$a_n$
$\sqrt{}$
$\sqrt[n]{}$
$\pi$
$\ln{}$
$\log$
$\theta$
$\sin{}$
$\cos{}$
$\tan{}$

$($
$)$
$[$
$]$
$\cap$
$\cup$
$,$
$\infty$