Question 7
Given a positive integer $n$. Show that for any positive real number $x$, we have the inequality:
$$
\frac{x^n (x^{n+1} + 1)}{x^n + 1} \le \left( \frac{x+1}{2} \right)^{2n+1}
$$
When does the equality take place?
$$
\frac{x^n (x^{n+1} + 1)}{x^n + 1} \le \left( \frac{x+1}{2} \right)^{2n+1}
$$
When does the equality take place?