<s>[INST] Given $x,y,z >0$ and $xy^2z^3 = 108 $, what is the minimum value of $x+y+z$? Given $x,y,z >0$ and $xy^2z^3 = 108 $, what is the minimum value of $x+y+z$ ?
This is a homework problem, so if someone could just give me an outline/hint of the method used to solve this it would be much appreciated. 
Thanks
 [/INST] To prove the solution is $x=1,y=2,z=3$ we can write the constrained problem as 
$min f(y,z)$ where $f(y,z) = \frac{108}{y^2 z^3} + y + z$.
Solving for the minimum of $f(y,z)$ we get:
$$
\frac{\partial f}{\partial y} = -2 \frac{108}{y^3 z^3} +1 = 0
$$
$$
(yz)^3 = 216
$$
$$
yz = 6
$$
$$
\frac{\partial f}{\partial z} = -3 \frac{108}{y^2 z^4} +1 = 0
$$
$$
(yz^2)^2 = 324
$$
$$
yz^2 =18
$$
Solving the pair of quadratic equations we get $y=2$ and $z=3$, and hence $x=1$. So the minimum value of the sum is $6$.
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