For positive real numbers $a,b$ prove that $\sqrt[3]{2(a+b)(\frac{1}{a}+\frac{1}{b})}\ge\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}$.
Solution 1:
Hint:
Let $x=\sqrt[3]{a/b}$, then you have to prove $$4+2x^3+{2\over x^3}\geq (x+{1\over x})^3$$
Notice that if $t>0$ then $$t^3-3t+2 = (t+2)(t-1)^2\geq 0 $$