How do you prove that ${x^y+y^x>1}$ if ${x,y>0}$ and both real? I saw a related problem on a Web puzzle page and it appears that the above statement is true when examined graphically in the unit square using a computer package like Mathematica. The result is obvious outside the unit square. I tried very hard to obtain an analytical proof (over several days) using my background in undergraduate mathematical analysis. Unfortunately I found the problem seems to be quite slippery - there are one or two bogus solutions on the Web that I found and discovered to be incorrect.
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https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1213001311 – Jean-Claude Arbaut Nov 28 '14 at 16:02
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Thanks Jean-Claud but your link seems to be to one of the bogus solutions I was talking about. – Paul Masham Nov 28 '14 at 16:57
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@PaulMasham what is wrong with the solution proposed by Aryabhatta at that link? – TZakrevskiy Nov 28 '14 at 17:38
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@TZakrevskiy It seemed to involve the Bernoulli inequality which is the wrong way round for x,y<1 in order to prove the result. However much of the script in the link was not visible to me so if I am mistaken please repeat the correct argument here or refer me to another link. Thanks. – Paul Masham Nov 28 '14 at 18:35