Test for a continuous function Let $f$ be a function defined in $[0, 6]$, continuous in $[0, 6]$
and it is provided of a third derivative in $]0, 6[.$ Which of the following assertions is false?
$$\fbox{A}\quad f \text{ has no asymptotes; }$$
$$\fbox{B}\quad f \text{ may have no critical points; }$$
$$\fbox{C}\quad f \text{ has a relative maximum or has a minimum
relative; }$$
$$\fbox{D}\quad f'' \text{ is continuous in } ]0; 6[;$$
$$\fbox{E}\quad \text{If } f'(5) = f''(5) = 0 \text{ and } f'''(5) = 7, \text{then } f \text{ has an inflection point with 
a horizontal tangent at } x = 5$$
Below there is the original question in Italian Language. Above there is the translation.


My attempt of resolution for to find the correct answer. The $\fbox{A}$ is true being $f$ is continuous in $[0,6]$.  The $\fbox{B}$ is true for the Weierstrass' theorem: remark that $[0,6]$ is closed set. If I think to the polynomial $\deg(p(x))=6$ and $\fbox{C}$ for me it is true. For the $\fbox{D}$ I have thought that if $f$ and it is provided of a third derivative in $]0,6[$, almost for $f''$ is continuous in $]0,6[$. I'd say the $\fbox{E}$ is false, but I can't justify it.
I ask if my reasoning is correct or there are incongruities.
 A: For me, C is false if you understand as a relative extremum (or local extremum) an extremum on a neighbourhood of a point in the interior of $[0,6]$. Indeed , here is a counterexample satisfying all the hypotheses, which has neither a local maximum nor a local minimum on $[0,6]$, albeit it has a maximum and a minimum:
$$f(x)=\frac 76(x-5)^3.$$
On the other hand, E is true, because if $f'''(5)=7$, it is positive in a small neighbourhood of $5$, say $I=(5-ε, 5+ε)$ (derivatives satisfy the  intermediate value property), so that $f''$ is increasing on this interval. Therefore , if $f''(5)=0$, we have $f''(x)<0$ on $(5-ε,5)$ and $f''(x)>0$ on $(5, 5+ε)$, so that $f'$ has a local  minimum on $I$, which corresponds to the definition of an inflection point.
A: Let's eliminate one by one.
A If $f$ is continuous in $[0,6]$ it means it is bounded for every $x \in [0,6]$. Therefore, no asymptotes. Thus, it's true.
B That is also true. For instance: $f(x) = e^x$. Remember that, in order to find critical points, we must take the first derivative and test for each $x$ that $f'=0$ or does not exist.
C That's also true. In order to find (local) maxima and minima we must consider both critical points and the bounds of the interval.
D Well, if $f'''$ is continuous in $]0,6[$ it means that $f''$ is differentiable in the same interval. Therefore, continuous as well.
E This is the false one. We may not ensure that the tangent will be horizontal. It would be correct if it said: ...$f'$ has a horizontal tangent at $x = 5$
