Is |f(b)−f(a)|>|b−a||f(b)-f(a)| > |b-a| true for f(x)=x+(1+ex)−1f(x)=x+(1+e^x)^{-1}?

I’d like to use this as part of a proof, but I couldn’t realize how to show this (and if it) is true. The function is: f(x)=x+(1+ex)1

Answer

f is differentiable at R, then by MVT,

f(a)f(b)=(ab)f(c)

with a<c<b.

but f(c)=1ec(1+ec)2=1+ec+e2c(1+ec)2

0<f(c)<1
thus |f(a)f(b)|<|ab|

your statement is not true for f.

Attribution
Source : Link , Question Author : user8011332 , Answer Author : hamam_Abdallah

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