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 $f$. [Figure] Fig. 1 (1a) [5 marks] Show that $x = e ^ { - 1 }$ and $x = e$ are the only zeros of $f$, and calculate the coordinates of the minimum point $T$ of $G _ { f }$. (for verification: $f ^ { \prime } ( x ) = \frac { 4 } { x } \cdot \ln x$ ) (1b) [6 marks] Show that $G _ { f }$ has exactly one inflection point $W$, and determine its coordinates and the equation of the tangent to $G _ { f }$ at the point $W$. (for verification: $x$-coordinate of $W : e$ ) (1c) [6 marks] Justify that $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x ) = - \infty$ and $\lim _ { x \rightarrow + \infty } f ^ { \prime } ( x ) = 0$ hold. Give $f ^ { \prime } ( 0,5 )$ and $f ^ { \prime } ( 10 )$ to one decimal place and draw the graph of the derivative function $f ^ { \prime }$ taking into account all previous results in Figure 1. (1d) [3 marks] Justify using Figure 1 that there are two values $c \in ] 0 ; 6 ]$ for which $\int _ { e ^ { - 1 } } ^ { c } f ( x ) \mathrm { dx } = 0$ holds. The rational function $h : x \mapsto 1,5 x - 4,5 + \frac { 1 } { x }$ with $x \in \mathbb { R } \backslash \{ 0 \}$ provides a good approximation for $f$ in a certain range. (1e) [2 marks] Specify the equations of the two asymptotes of the graph of $h$. (1f) [5 marks] In the fourth quadrant, $G _ { f }$ together with the $x$-axis and the lines with equations $x = 1$ and $x = 2$ enclose a region whose area is approximately 1.623. Determine the percentage deviation from this value if the function $h$ is used as an approximation for the function $f$ when calculating the area. By reflecting $G _ { f }$ across the line $x = 4$, the graph of a function $g$ defined on $] - \infty ; 8 [$ is created. This graph is denoted by $G _ { g }$. (2a) [2 marks] Draw $G _ { g }$ in Figure 1. (2b) [3 marks] The described reflection of $G _ { f }$ across the line $x = 4$ can be replaced by a reflection of $G _ { f }$ across the $y$-axis followed by a translation. Describe this translation and specify $a , b \in \mathbb { R }$ such that $g ( x ) = f ( a x + b )$ for $x \in ] - \infty ; 8 [$. In the following, the ``w-shaped'' curve $k$ is considered, which consists of the part of $G _ { f }$ restricted to $0,2 \leq x \leq 4$ and the part of $G _ { g }$ restricted to $4 < x \leq 7,8$. The curve $k$ is translated 12 units in the negative $z$-direction. The area swept out in this process serves as a model for a 12-meter-long aquarium, which is completed by two flat walls at the front and back to form a basin (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality. [Figure] Fig. 2 (2c) [3 marks] The aquarium walls form a tunnel at the bottom through which visitors can walk. Calculate the size of the angle that the left and right tunnel walls enclose with each other. The aquarium is completely filled with water. (2d) [2 marks] Calculate the maximum water depth of the aquarium. \footnotetext{(c) \href{http://Abiturloesung.de}{Abiturloesung.de} } (2e) [3 marks] The volume of water in the aquarium can be calculated analogously to the volume of a prism with base area $G$ and height $h$. Explain that the term $24 \cdot \int _ { 0,2 } ^ { 4 } ( f ( 0,2 ) - f ( x ) ) \mathrm { dx }$ describes the water volume in the completely filled aquarium in cubic meters.
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}
\textbf{(1a)} [5 marks] Show that $x = e ^ { - 1 }$ and $x = e$ are the only zeros of $f$, and calculate the coordinates of the minimum point $T$ of $G _ { f }$.\\
(for verification: $f ^ { \prime } ( x ) = \frac { 4 } { x } \cdot \ln x$ )
\textbf{(1b)} [6 marks] Show that $G _ { f }$ has exactly one inflection point $W$, and determine its coordinates and the equation of the tangent to $G _ { f }$ at the point $W$.\\
(for verification: $x$-coordinate of $W : e$ )
\textbf{(1c)} [6 marks] Justify that $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x ) = - \infty$ and $\lim _ { x \rightarrow + \infty } f ^ { \prime } ( x ) = 0$ hold. Give $f ^ { \prime } ( 0,5 )$ and $f ^ { \prime } ( 10 )$ to one decimal place and draw the graph of the derivative function $f ^ { \prime }$ taking into account all previous results in Figure 1.
\textbf{(1d)} [3 marks] Justify using Figure 1 that there are two values $c \in ] 0 ; 6 ]$ for which $\int _ { e ^ { - 1 } } ^ { c } f ( x ) \mathrm { dx } = 0$ holds.
The rational function $h : x \mapsto 1,5 x - 4,5 + \frac { 1 } { x }$ with $x \in \mathbb { R } \backslash \{ 0 \}$ provides a good approximation for $f$ in a certain range.
\textbf{(1e)} [2 marks] Specify the equations of the two asymptotes of the graph of $h$.
\textbf{(1f)} [5 marks] In the fourth quadrant, $G _ { f }$ together with the $x$-axis and the lines with equations $x = 1$ and $x = 2$ enclose a region whose area is approximately 1.623. Determine the percentage deviation from this value if the function $h$ is used as an approximation for the function $f$ when calculating the area.
By reflecting $G _ { f }$ across the line $x = 4$, the graph of a function $g$ defined on $] - \infty ; 8 [$ is created. This graph is denoted by $G _ { g }$.\\
\textbf{(2a)} [2 marks] Draw $G _ { g }$ in Figure 1.
\textbf{(2b)} [3 marks] The described reflection of $G _ { f }$ across the line $x = 4$ can be replaced by a reflection of $G _ { f }$ across the $y$-axis followed by a translation. Describe this translation and specify $a , b \in \mathbb { R }$ such that $g ( x ) = f ( a x + b )$ for $x \in ] - \infty ; 8 [$.
In the following, the ``w-shaped'' curve $k$ is considered, which consists of the part of $G _ { f }$ restricted to $0,2 \leq x \leq 4$ and the part of $G _ { g }$ restricted to $4 < x \leq 7,8$. The curve $k$ is translated 12 units in the negative $z$-direction. The area swept out in this process serves as a model for a 12-meter-long aquarium, which is completed by two flat walls at the front and back to form a basin (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality.
\begin{figure}[h]
\begin{center}
\textit{[Figure]}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
\textbf{(2c)} [3 marks] The aquarium walls form a tunnel at the bottom through which visitors can walk. Calculate the size of the angle that the left and right tunnel walls enclose with each other.
The aquarium is completely filled with water.
\textbf{(2d)} [2 marks] Calculate the maximum water depth of the aquarium.
\footnotetext{(c) \href{http://Abiturloesung.de}{Abiturloesung.de}
}
\textbf{(2e)} [3 marks] The volume of water in the aquarium can be calculated analogously to the volume of a prism with base area $G$ and height $h$.\\
Explain that the term $24 \cdot \int _ { 0,2 } ^ { 4 } ( f ( 0,2 ) - f ( x ) ) \mathrm { dx }$ describes the water volume in the completely filled aquarium in cubic meters.