\[ A = \frac{(x + \frac{1}{x})^6 - (x^3 + \frac{1}{x^3})^2}{(x + \frac{1}{x})^3 + (x^3 + \frac{1}{x^3} )} \]\[ A = (x + \frac{1}{x})^3 - (x^3 + \frac{1}{x^3}) = 3(x + \frac{1}{x}) \ge 6 \]

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.