Help!

The expression has a minimum value of 6, when x = 1.

$\frac{\left( x + \dfrac{1}{x} \right)^6 - \left( x^6 + \dfrac{1}{x^6} \right) - 2}{\left( x + \dfrac{1}{x} \right)^3 + \left( x^3 + \dfrac{1}{x^3} \right)}$

Just by inspection I can see that as x moves away from 1 in either direction the function value will increase.

So the minimum will occur when x=1

that minimum will be

$\frac{2^6-2-2}{2^3+2}=\frac{60}{10}=6$

Just as our guest said,