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+\frac1x` where
`x>0`.

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


`f'(x)=1-\frac1{x^2}=0\implies1=\frac1{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`.