Proving a point lies between two lines on a graph

Question

Prove that the point $(x,\frac{1}{\sqrt{x+2}-\sqrt{x}})$ lies between the two curves: $y=\sqrt{x}$ and $y=\sqrt{x+2}$.

My initial idea was: $\sqrt{x} < \frac{1}{\sqrt{x+2}-\sqrt{x}} < \sqrt{x+2}$. Then I rationalised the middle expression. Then i just got stuck.

Answer 1

As you have mentioned, \begin{align*} \frac{1}{\sqrt{x+2}-\sqrt{x}} = \frac{1}{\sqrt{x+2} - \sqrt{x}}\times\frac{\sqrt{x+2} + \sqrt{x}}{\sqrt{x+2} + \sqrt{x}} = \frac{\sqrt{x+2} + \sqrt{x}}{2} \end{align*} which is the arithmetic mean between $\sqrt{x}$ and $\sqrt{x+2}$. More precisely, the claim results from noticing that \begin{align*} \frac{\sqrt{x+2}+\sqrt{x}}{2} - \sqrt{x} = \frac{\sqrt{x+2}-\sqrt{x}}{2} > 0 \end{align*} as well as \begin{align*} \sqrt{x+2} - \frac{\sqrt{x+2}+\sqrt{x}}{2} = \frac{\sqrt{x+2} - \sqrt{x}}{2} > 0 \end{align*}