$\Big| \dfrac{df}{dx}(x)\Big|\leq 5$ for all $x$ 
$f:\mathbb{R}\rightarrow \mathbb{R}$ is such that $f(0)=0$ and $\Big| \dfrac{df}{dx}(x)\Big|\leq 5$ for all $x$. We can conclude that $f(1)$ is in

*

*$(5,6)$


*$[-5,5]$


*$(-\infty,-5)\cup (5,\infty)$


*$[-4,4]$

The answer would be 2. (You can also find a solution here and here and here)
My question is: Is there an example of $f$ for which $f(1)$ can be $5$ or $-5$?
For example if we take $f(x)=5\sin x$, $f(1)$ is not $5$(obviously), but we can not take $f(x)=5\sin x +c$, where $c\neq 0$, since $f(0)=0$.
Can someone help me here? I am very bad at finding examples. I guess someone has to start with choosing $f(0)=0$ and $f(1)=5$, but what would be the next steps? Thanks.
Added:
So stupid, thanks to Kenny Lau, I understood that. What about if I add an extra condition that $f$ has non-constant derivative.

Mean value Theorem:
$\dfrac{f(1) -f(0)}{1-0} = f'(t)$, with $t \in (0,1).$
Hence:
$|f(1)| = |f'(t)| \le 5.$
$\Rightarrow: f(1) \in [-5,5].$
