# 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: $f(x)=x+(1+e^x)^{-1}$

$f$ is differentiable at $\mathbb R$, then by MVT,

with $a .

but

thus

your statement is not true for $f$.