(1) [5 marks] Given is the function $f : x \mapsto \frac { e ^ { 2 x } } { x }$ with domain $D _ { f } = \mathbb { R } \backslash \{ 0 \}$. Determine the location and type of the extremum point of the graph of $f$. Given is the function $f : x \mapsto 1 - \frac { 1 } { x ^ { 2 } }$ defined in $\mathbb { R } \backslash \{ 0 \}$, which has zeros $x _ { 1 } = - 1$ and $x _ { 2 } = 1$. Figure 1 shows the graph of $f$, which is symmetric with respect to the y-axis. Furthermore, the line $g$ with equation $y = - 3$ is given. [Figure] (2a) [1 marks] Show that one of the points where $g$ intersects the graph of $f$ has the x-coordinate $\frac { 1 } { 2 }$. (2b) [4 marks] Determine by calculation the area enclosed by the graph of $f$, the x-axis, and the line $g$. The adjacent Figure 2 shows the graph of a function $f$. [Figure] (3a) [3 marks] One of the following graphs I, II, and III belongs to the first derivative function of $f$. Specify this graph. Justify why the other two graphs are not suitable. [Figure][Figure][Figure] (3b) [2 marks] The function $F$ is an antiderivative of $f$. Specify the monotonicity behavior of $F$ on the interval $[ 1 ; 3 ]$. Justify your statement. )$} Consider a family of functions $h _ { k }$ with $k \in \mathbb { R } ^ { + }$, which differ only in their respective domains $D _ { k }$. It holds that $h _ { k } : x \mapsto \cos x$ with $D _ { k } = [ 0 ; k ]$. Figure 4 shows the graph of the function $h _ { 7 }$. Specify the largest possible value of $k$ such that the corresponding function $h _ { k }$ is invertible. For this value of $k$, sketch the graph of the inverse function of $h _ { k }$ in Figure 4 and pay particular attention to the intersection point of the graphs of the function and its inverse. [Figure] (4b) [2 marks] Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point. Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.
\textbf{(1)} [5 marks] Given is the function $f : x \mapsto \frac { e ^ { 2 x } } { x }$ with domain $D _ { f } = \mathbb { R } \backslash \{ 0 \}$.\\
Determine the location and type of the extremum point of the graph of $f$.
Given is the function $f : x \mapsto 1 - \frac { 1 } { x ^ { 2 } }$ defined in $\mathbb { R } \backslash \{ 0 \}$, which has zeros $x _ { 1 } = - 1$ and $x _ { 2 } = 1$. Figure 1 shows the graph of $f$, which is symmetric with respect to the y-axis. Furthermore, the line $g$ with equation $y = - 3$ is given.\\
\textit{[Figure]}
\textbf{(2a)} [1 marks] Show that one of the points where $g$ intersects the graph of $f$ has the x-coordinate $\frac { 1 } { 2 }$.
\textbf{(2b)} [4 marks] Determine by calculation the area enclosed by the graph of $f$, the x-axis, and the line $g$.
The adjacent Figure 2 shows the graph of a function $f$.\\
\textit{[Figure]}
\textbf{(3a)} [3 marks] One of the following graphs I, II, and III belongs to the first derivative function of $f$. Specify this graph. Justify why the other two graphs are not suitable.\\
\textit{[Figure]}\\
\textit{[Figure]}\\
\textit{[Figure]}
\textbf{(3b)} [2 marks] The function $F$ is an antiderivative of $f$. Specify the monotonicity behavior of $F$ on the interval $[ 1 ; 3 ]$. Justify your statement.
)$}
Consider a family of functions $h _ { k }$ with $k \in \mathbb { R } ^ { + }$, which differ only in their respective domains $D _ { k }$.\\
It holds that $h _ { k } : x \mapsto \cos x$ with $D _ { k } = [ 0 ; k ]$.\\
Figure 4 shows the graph of the function $h _ { 7 }$. Specify the largest possible value of $k$ such that the corresponding function $h _ { k }$ is invertible. For this value of $k$, sketch the graph of the inverse function of $h _ { k }$ in Figure 4 and pay particular attention to the intersection point of the graphs of the function and its inverse.\\
\textit{[Figure]}
\textbf{(4b)} [2 marks] Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.
Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.