\textbf{(1)} [4 marks] For each of the functions $f _ { 1 }$ and $f _ { 2 }$, specify the maximum domain and the zero.

$$f _ { 1 } : x \mapsto \frac { 2 x + 3 } { x ^ { 2 } - 4 } \quad f _ { 2 } : x \mapsto \ln ( x + 2 )$$

\textbf{(2)} [3 marks] Specify the term of a function defined on $\mathbb { R }$ whose graph has a horizontal tangent at the point (2|1), but no extremum.

\textbf{(3)} [5 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = - x ^ { 3 } + 9 x ^ { 2 } - 15 x - 25$.\\
Prove that $f$ has the following properties:\\
(1) The graph of $f$ has slope $-15$ at the point $x = 0$.\\
(2) The graph of $f$ has the $x$-axis as a tangent at the point $A ( 5 \mid f ( 5 ) )$.\\
(3) The tangent $t$ to the graph of the function $f$ at the point $B ( - 1 \mid f ( - 1 ) )$ can be described by the equation $y = - 36 x - 36$.

\textbf{(4)} [3 marks] The figure shows the graph $G _ { f }$ of a function $f$ defined on $\mathbb { R }$ with the inflection point $W ( 1 \mid 4 )$.

Using the figure, determine approximately the value of the derivative of $f$ at the point $x = 1$.

Sketch the graph of the derivative function $f ^ { \prime }$ of $f$ into the figure; in doing so, pay particular attention to the location of the zeros of $f ^ { \prime }$ and the approximate value determined for $f ^ { \prime } ( 1 )$.\\
\textit{[Figure]}

For each value of $a$ with $a \in \mathbb { R } ^ { + }$, a function $f _ { a }$ is given by $f _ { a } ( x ) = \frac { 1 } { a } \cdot x ^ { 3 } - x$ with $x \in \mathbb { R }$.

\textbf{(5a)} [2 marks] One of the two figures shows a graph of $f _ { a }$. Specify which figure this is. Justify your answer.

\begin{figure}[h]
\begin{center}
  \textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}

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

\textbf{(5b)} [3 marks] For each value of $a$, the graph of $f _ { a }$ has exactly two extrema. Determine the value of $a$ for which the graph of the function $f _ { a }$ has an extremum at the point $x = 3$.

Given is the function $f : x \mapsto 2 \cdot \left( ( \ln x ) ^ { 2 } - 1 \right)$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph $G _ { f }$ of $f$.

\begin{figure}[h]
\begin{center}
  \textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}