Prove that if \(\displaystyle{e}{ < }{y}{

Dumaen80p3

Dumaen80p3

Answered question

2022-03-30

Prove that if e<y<x then xy<yx
My try:
I tried to use Taylor's theorem:
yxxy=exlnyeylnx=1+xlny+o(xlny)1ylnxo(ylnx)=
=lnyxlnxy+o(xlny)o(ylnx)=lnyxxy+o(xlny)o(ylnx)
Hovewer I have a problem to show that yxxy>0 because I can't say that lnyxxy>0 because then I use with theses.
Have you some idea?

Answer & Explanation

diocedss33

diocedss33

Beginner2022-03-31Added 12 answers

exln(y)<eyln(x) is equivalent to ln(x)x<ln(y)y
Let f(x)=ln(x)x,f'(x)=1ln(x)x2<0 if x>e, so if e<y<x,
f(x)<f(y)

diocedss33

diocedss33

Beginner2022-04-01Added 12 answers

The hint:
Prove that f(x)=xlnx increases for x>e.

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