<s>[INST] Picking out unique solution of $x,y,z$ from two constraints 
Let $x,y,z$ be positive real numbers in $\mathbb{R}$ satisfying the conditions:$x+y+z=12$ and $x^3 y^4z^5=(.1)600^3$. Then the value of $x^3 + y^3 +z^3=?$

This is a question from JEE mains, in it I have a doubt of how we are able to uniquely determine an $x,y,z$ triplet from just two equations(upto permutation)! The answer involves observing $x=3, y=4 $ and $z=5$ but how can we make that inference?
 [/INST] For positive reals, we can use AM-GM inequality. This inequality for $12$ terms applies as
$$\frac{3\cdot (x/3)+4\cdot (y/4)+5\cdot (z/5)}{3+4+5} \ge \left(\frac{x^3}{3^3}\cdot \frac{y^4}{4^4}\cdot \frac{z^5}{5^5} \right)^{1/(3+4+5)}$$
$$\Rightarrow x^3\cdot y^4 \cdot z^5 \le 3^3 \cdot 2^8 \cdot 5^5$$
$$\Rightarrow x^3y^4z^5 \le (0.1)(600)^3$$
where equality (maximum) is achieved when each term is equal i.e.,
$$\frac{x}{3}=\frac{y}{4}=\frac{z}{5}=\frac{12}{12}=1$$
from which one infers $x=3, y=4, z=5$.
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