I dont get why $\text{(I)}$ is not correct. The slope at $x=3$ is positive, meaning that $f'(x)>0$. Can anyone explain? Correct answer is $\text{(C)}$.

---

\begin{array}{|c|c|c|}\hline \;\;\;\;\;x\;\;\;\;\; & \;\;\;\;\;0\;\;\;\;\;\ & \;\;\;\;\;2\;\;\;\;\;\ & \;\;\;\;\;4\;\;\;\;\;\ & \;\;\;\;\;8\;\;\;\;\;\ \\
\hline f(x) & 3 & 4 & 9 & 13\\
\hline f'(x) & 0 & 1 & 1 & 2\\ \hline \end{array}
The table above gives values of a differentiable function $f$ and its derivative at selected values of $x$. If $h$ is the function given by $h(x)=f(2x)$, which of the following statements must be true?

$\text{(I)}$ $h$ is increasing on $2\lt x \lt4$  

$\text{(II)}$ There exists $c$, where $0\lt c\lt4$, such that $h(c)=12$.  

$\text{(III)}$ There exists $c$, where $0\lt c\lt2$, such that $h'(c)=3$.

$\text{(A)}$ $\text{II}$ only  

$\text{(B)}$ $\text{I}$ and $\text{III}$ only  

$\text{(C)}$ $\text{II}$ and $\text{III}$ only  

$\text{(D)}$ $\text{I}$, $\text{II}$, and $\text{III}$