AHSME 1979 Problem 29

For each positive number $x$, let $f(x)=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2} {\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$. The minimum value of $f(x)$ is $\textbf{(A) }1 \textbf{(B) }2 \textbf{(C) }3 \textbf{(D) }4 \textbf{(E) }6$