2. $\displaystyle \begin{gathered} \left( {a - b} \right)^2 \geqslant 0 \Rightarrow \quad a^2 + b^2 \geqslant 2ab \hfill \\ \text{let }a = \sqrt x \,\& \,b = \frac{1} {{\sqrt x }} \hfill \\ \end{gathered}$
3. And since you're in calculus, you can also differentiate the function $\displaystyle f(x)=x+\frac 1x$