Rolle theorem on infinite interval We have :
$f(x)$ is continuous  on $[1,\infty]$ and differentiable on $(1,\infty)$
$\lim\limits_{x \to \infty}f(x) = f(1)$
we have to prove that : there is $b\in(1,\infty)$ such that $f'(b) = 0$
I'm sure we have to use Rolle's theorem so, I tried using Mean Value theorem and using the limit definition  at $\infty$
Any ideas how can I use them ?
Update :
after seeking the answers that I've got :
I'm having trouble finding $\boldsymbol x_{\boldsymbol 1}\neq\boldsymbol x_{\boldsymbol 2}$ such that $\boldsymbol{f(x}_{\boldsymbol 1}\boldsymbol {)=f(x}_{\boldsymbol 2}\boldsymbol )$
*I need a formal solution

Outline:
You may, and should, assume that $f$ is non-constant.
Pick $a$ so that $f(a)\ne f(1)$ (if no such $a$ exists $f$ is a constant function). 
Choose a value $c$ that is in between $f(1)$ and $f(a)$.  Use the Intermediate Value Theorem to show there is a $x_1\in(1,a)$ with $f(x_1)=c$. 
Use the Intermediate Value Theorem and your limit condition to show there is  an $x_2\in(a,\infty)$ with $ f(x_2)=c$. 
Now you're set to apply Rolle's Theorem.
