Question

How do you find a positive number such that the sum of the number and its reciprocal is as small as possible?

Answer 1

The smallest sum of a number `n` and its reciprocal `1/n` is `2` which occurs when `n = 1`. Any other value of `n` will produce a larger sum.

Explanation:

Let us consider a positive number `n`, making sure `n \ne 0` so that we don't have an undefined reciprocal.

We want to find a `1/n` such that `n + 1/n` is minimized. We can call this sum a function `f(n) = n + 1/n`.

Now we take the derivative of `f(n)` w.r.t. `n` and set it equal to zero to obtain the minimum.

`f'(n) = 1 -1/n^2`

`1 - 1/n^2 = 0`
`1 = 1/n^2`
`n^2 = 1`
`n = +- 1`

However, we reject the negative value as `n > 0`. Hence, `n = 1`.

So the minimum sum obtainable is `f(1) = 1+ 1/1 = 2`

Hence, the smallest sum of a number `n` and its reciprocal `1/n` is 2 when `n = 1`. Any other value of `n` will produce a larger sum.