Find a positive number x such that the sum of 9x and 1/x is as small as possible. Does this problem require optimization over an open interval or a closed interval?

Question

Find a positive number x such that the sum of 9x and 1/x is as small as possible.

x = ?

Does this problem require optimization over an open interval or a closed interval?

Answer 1

f(x) = 9x + 1/x

f'(x) = 9 - 1/x² = (9x² - 1) / x²

f'(x) = 0 when 9x² = 1. So x = 1/3

Answer 2

S(x) = 9x + (1/x)

S(x) = 9x² + 1) / x

Differenctiate using quotient rule.