<s>[INST] What is the relationship between $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3$ and $x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$? Is $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3$ > or < or = $x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$, given $x_1$, $x_2$ and $x_3$ are all positive?
I know there exists a real number $a\in(1,2)$ such that $\frac{1}{3}x=x^{\frac {1}{3}}$, but there are three variables even though they are all positive.
Also I know how to simplify $\frac{1}{3}x_1+\frac{1}{3}x_2+\frac{1}{3}x_3-x_1^{\frac{1}{3}}x_2^{\frac{1}{3}}x_3^{\frac{1}{3}}$, but then I get stuck too.
I also tried to relate this to the roots of polynomial, but I don't quite know how.
Could anyone help please?
 [/INST] As you said, $x_1$,$x_2$,$x_3$ are positive reals. Furthermore,  by AM-GM inequality $\frac{x_1 + x_2 + x_3}{3} \geq  (x_1)^\frac{1}{3}(x_2)^\frac{1}{3}(x_3)^\frac{1}{3}$
so the proof is using just AM-GM inequality.
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