f is a second-differentiable function at $\displaystyle (0,\infty) $ so $\displaystyle f''(x)>0$ to every $\displaystyle x\in(0,\infty)$

i need to prove that if:$\displaystyle lim{}_{x\rightarrow\infty}f(x)=\ell$ $\displaystyle (\ell $ is finite), so -

(1)$\displaystyle f'(x)<0 $ to every $\displaystyle x\in(0,\infty)$

(2)$\displaystyle sup\: f'((0,\infty))=0 $

(3)$\displaystyle lim{}_{x\rightarrow\infty}f'(x)=0 $