perplexus.info :: Just Math : A simple inequality

A simple inequality (Posted on 2014-12-31)

Given:
x>0, y>0, x+y=1

Prove that
(1 + 1/x)(1 + 1/y) ≥ 9.

No Solution Yet Submitted by Ady TZIDON

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re: Solution

Comment 4 of 4 |

(In reply to Solution by Bractals)

Another way

((x + 1)/x)((y+1)/y) >= 9

xy + x + y + 1 >= 9xy

But x + y = 1,

So we need 2 >= 8xy

or 1/4 >= xy

By symmetry, xy has its maximum when x = y = 1/2,

so the maximum value of xy is 1/4.

q.e.d.

Posted by Steve Herman on 2014-12-31 15:58:47

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