Given is the function $f : x \mapsto \ln ( x - 3 )$ with maximal domain $D$ and derivative function $f ^ { \prime }$.\\

\textbf{(1a)} [2 marks] State $D$ and the zero of $f$.\\

\textbf{(1b)} [3 marks] Determine the point $x \in D$ for which $f ^ { \prime } ( x ) = 2$ holds.

Given is the function $g : x \mapsto \frac { 1 } { x ^ { 2 } } - 1$ defined in $\mathbb { R } \backslash \{ 0 \}$.\\

\textbf{(2a)} [2 marks] State an equation of the horizontal asymptote of the graph of $g$ and the range of $g$.

\textbf{(2b)} [3 marks] Calculate the value of the integral $\int _ { \frac { 1 } { 2 } } ^ { 2 } g ( x ) \mathrm { dx }$.

A polynomial function $f$ defined in $\mathbb { R }$, which is not linear, with first derivative function $f ^ { \prime }$ and second derivative function $f ^ { \prime \prime }$ has the following properties:

\begin{itemize}
  \item $f$ has a zero at $x _ { 1 }$.
  \item It holds that $f ^ { \prime } \left( x _ { 2 } \right) = 0$ and $f ^ { \prime \prime } \left( x _ { 2 } \right) \neq 0$.
  \item $f ^ { \prime }$ has a local minimum at the point $x _ { 3 }$.
\end{itemize}

Figure 1 shows the positions of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$.\\
\textit{[Figure]}

\textbf{(3a)} [2 marks] Justify that the degree of $f$ is at least 3.

\textbf{(3b)} [3 marks] Sketch a possible graph of $f$ in Figure 1.

Figure 2 shows the graph of the function $g$ defined in $\mathbb { R }$, whose only extreme points are $( - 1 \mid 1 )$ and $( 0 \mid 0 )$, as well as the point $P$.

\begin{figure}[h]
\begin{center}
  \textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

\textbf{(4a)} [2 marks] State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.

Subtask Part A 4b $( 3 \mathrm { marks } )$\\
The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.

Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.

\begin{figure}[h]
\begin{center}
  \textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}