<s>[INST] Minimum value of the function $\sqrt{(1+1/m)(1+1/n)}$ If $m, n$ are positive real variables whose sum is a constant $k$, then what is the minimum value of $$\sqrt{\bigg(1 + \frac{1}{m}\bigg)\bigg(1 + \frac{1}{n}\bigg)}$$
 [/INST] By the Cauchy-Schwarz Inequality, we have:
$$\left(1^2 + \left(\frac{1}{\sqrt{m}}\right)^2\right) \left(1^2 + \left(\frac{1}{\sqrt{n}}\right)^2\right) \ge \left(1 + \frac{1}{\sqrt{mn}}\right)^2\\
\sqrt{\left(1 + \frac{1}{m}\right)\left(1 + \frac{1}{n}\right)} \ge 1 + \frac{1}{\sqrt{mn}}$$
But we also know by the AM-GM inequality that
$$\sqrt{mn} \le \frac{m + n}{2}$$
or
$$\frac{1}{\sqrt{mn}} \ge \frac{2}{m + n}$$
So this gives us 
$$\begin{align}\sqrt{\left(1 + \frac{1}{m}\right)\left(1 + \frac{1}{n}\right)} &\ge 1 + \frac{2}{m + n}\\
&= 1 + \frac{2}{k}\end{align}$$
with equality at $m = n = \frac{k}{2}$.
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