"Find a positive number such that the sum of the number and its reciprocal is as small as possible."
??
$\displaystyle S = x + \frac{1}{x}$.
Find the minimum using calculus:
$\displaystyle \frac{dS}{dx} = 1 - \frac{1}{x^2}$.
Stationary points: Solve $\displaystyle 0 = 1 - \frac{1}{x^2}$ subject to the restriction x > 0.
I get $\displaystyle x = 1$. Test nature ....
The minimum value occurs at x = 1 and the minimum value is S = 2.