Show that if 1>x>0, then x−1>=ln(x)>=1−1/x

Madilyn Quinn 2022-10-16 Answered
Show that if 1 > x > 0, then x 1 ln ( x ) 1 1 x
I know how to prove it using the MVT and I can prove it for x > 1 but I don't understand how to prove it for x > 0
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Answers (2)

SoroAlcommai9
Answered 2022-10-17 Author has 13 answers
For 0 < x < 1
ln x = x 1 d t t .
Since
1 1 t 1 x ,
we have,
x 1 d t ln x x 1 1 x d t ,
and
1 x ln x 1 x x .
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Ignacio Riggs
Answered 2022-10-18 Author has 4 answers
Well, all functions are equal at x = 1. What can you say about the relative size of their derivatives, 1, 1 x , and 1 x 2 when 0 < x < 1? What can you conclude about the functions?
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