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 \(f'(x)<0\) for \(x<1\), and \(f'(x)>0\) for \(x>1\), which means a minimum occurs at \(x=1\).
