Find the minimum value of this rational expression

Question

A question asks, "If x is a postive real number, then find the minimum value of $[{x + (1/x)}^6 - {x^6 + (1/x)^6} - 2]/ [{x + (1/x)}^3 + {x^3 + (1/x)^3}]$

Question Image

My try:- I expanded and simplified the numerator and then applied AM - GM and got Numerator $>= 12*sqrt(10) + 18$. Then applying AM-GM on denominator, we get denominator $>= 4*sqrt(6)$.

But I do not know how to combine these two results. How can I proceed?

Answer 1

Hint: It simplifies to $$3(x+\frac{1}{x})$$, now use that $$a+\frac{1}{a}\geq 2$$ for $a>0$