Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$
The local minimum occurs at x=1, but there's a local maximum at x=-1. The question has missing details. It may be better expressed as, find a minimum of A, for x >0.