BMAD Posted October 10, 2017 Report Share Posted October 10, 2017 Prove that x^(1/x) = x^-x has only one solution Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted October 10, 2017 Report Share Posted October 10, 2017 Maybe, maybe not. Spoiler Taking the natural log of both sides yields (1/x) ln(x) = (-x) ln(x) ln(x) [1/x + x] = 0 This has solutions ln(x) = 0 ==> x = 1 And 1/x + x = 0 ==> x^{2} + 1 = 0 ==> x = (+/-) i So there are three solutions, one of which is real. Quote Link to comment Share on other sites More sharing options...

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## BMAD

Prove that x^(1/x) = x^-x has only one solution

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