\(f(n) = 1 + xln(x + \sqrt{x^2 + 1}) - \sqrt{1 + x^2}\)
\(f'(x) = ln (x + \sqrt{x^2 +1}) + \frac x{x + \sqrt{x^2 + 1}} \times \left[1 + \frac x{\sqrt{x^2 + 1}}\right] - \frac x{\sqrt{1 + x^2}}\)
\(= ln (x + \sqrt{x^2 +1}) + \frac x{ \sqrt{x^2 + 1}} - \frac x{\sqrt{1 + x^2}}\)
\(= ln (x + \sqrt{x^2 +1}) \)
\(f'(x) = ln (x + \sqrt{x^2 + 1})\) \(x \ge 0\)
\(f'(x) \ge 0\)
f(x) is an \(f^n \;\forall \;x \ge 0\)
⇒ \(\;\forall \;x \ge 0\)
\(f(x) \ge f(0)\)
⇒ \(1 + x ln(x + \sqrt{x^2 + 1}) - \sqrt{1 + x^2} \ge 0\)
⇒ \(1 + x ln(x + \sqrt{x^2 + 1}) \ge \sqrt{1 + x^2}\) \(\;\forall \;x \ge 0\)
