If x, y are positive real numbers such that x + y = 1, prove that (1 + 1/x)(1+1/y)≥9.

Question

If x, y are positive real numbers such that x + y = 1, prove that \(\big(1+\frac1x\big)\big(1+\frac1y\big)\) > 9.

Answer 1

For positive real numbers, x, y applying AM > GM, we have

\(\frac{x+y}{2}\) > \(\sqrt{xy}\)

⇒ \(\frac12\)  > \(\sqrt{xy}\)         (∵ x + y = 1)

⇒ 1 ≥ 2\(\sqrt{xy}\)         ⇒ 1 > 4xy ⇒ 2 > 8xy ⇒ 1 + 1 > 8xy

⇒ 1 + \(x\) + y > 8xy ⇒ 1 + \(x\) + y + xy > 9xy ⇒ (1 + \(x\)) (1 + y) > 9xy

⇒ \(\frac{(1+x)(1+y)}{xy}\) > 9  ⇒   \(\big(\frac{x+1}{x}\big)\big(\frac{y+1}{y}\big)\) > 9 ⇒  \(\big(1+\frac1x\big)\big(1+\frac1y\big)\) > 9