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

1 Answer
Mar 14, 2018

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.