Question

Find a positive number such that the sum of the number and its reciprocal is as small as possible?

Answer 1

You're minimizing the function $f(x)=x+\dfrac1x$ where $x>0$.

Take the derivative, set to zero and find the critical points:

$f'(x)=1-\dfrac1{x^2}=0\implies1=\dfrac1{x^2}$

This has two roots, $x=\pm1$, but we ignore the negative root.

Check the signs of the derivative to the left and right of $x=1$ and you'll see that $f'(x)$ for $x$, and $f'(x)>0$ for $x>1$, which means a minimum occurs at $x=1$.