10. Determine the maximum value attained by
$$
\frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1}
$$
over real numbers $x>1$.
Answer: $\frac{1}{6}$
Solution: We have the following algebra:
$$
\begin{aligned}
\frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1} & =\frac{x-\frac{1}{x}}{x^{3}+2-\frac{1}{x^{3}}} \\
& =\frac{x-\frac{1}{x}}{\left(x-\frac{1}{x}\right)^{3}+2+3\left(x-\frac{1}{x}\right)} \\
& \leq \frac{x-\frac{1}{x}}{3\left(x-\frac{1}{x}\right)+3\left(x-\frac{1}{x}\right)}=\frac{1}{6}
\end{aligned}
$$
where $\left(x-\frac{1}{x}\right)^{3}+1+1 \geq 3\left(x-\frac{1}{x}\right)$ in the denominator was deduced by the AM-GM inequality. As a quick check, equality holds where $x-\frac{1}{x}=1$ or when $x=\frac{1+\sqrt{5}}{2}$.