second-differentiable function proving help

[orir]

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 \(\displaystyle x\in(0,\infty)\)

 

[pka]

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 \(\displaystyle x\in(0,\infty)\)

From the given, you know that the function is bounded.
You know that the derivative is increasing everywhere.
You know that the graph is concave up everywhere.

If we suppose that \(\displaystyle \exists c>0\) such that \(\displaystyle f'(c)\ge 0\), how does that contradict the given?