Hello everybody! I am new here and probably going to visit here pretty often in the future. Sometimes I do not get very clear answer from the lecturers so I hope to get more sensible answers from PF.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

I need to show that x^y > y^x whenever y > x ≥ e.

3. The attempt at a solution

At first I start by multiplying by ln(): y*ln(x) > x*ln(y)

Then I take g(x,y) such that g(x,y) = y*ln(x) - x*ln(y) and 2nd order derivatives and show det(H(g)) < 0 whenever y > x ≥ e

My purpose with this is to show that there are no real local or global critical points in g(x,y) when y > x >= e, and conclude that g(x) = x^y - y^x diverges. I was not told why I can not use Hessian determinant to make this conclusion.

Why I can not use Hessian approach to prove this? What would be the correct way to do it? Can I prove it with induction?

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# Homework Help: Prove that x^y > y^x when y>x>e

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