Proving or disproving: If 0<a<b<1, then (1−a)^b>(1−b)^a

Franco Franco

Franco Franco

Answered question

2022-01-22

Proving or disproving: If 0<a<b<1, then (1a)b>(1b)a

Answer & Explanation

sineurosi0f

sineurosi0f

Beginner2022-01-23Added 14 answers

Let f(x)=ln(1x)x.
Thus, we need to prove that f decreases, which is not so obvious without derivative.
We have:
f(x)=ln(1x)+x1xx2
Now, let g(x)=ln(1x)+x1x.
Thus,
g(x)=1(1x)211x=x(1x)2>0,
which says
g(x)>g(0)=0
and we got that f decreases.

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