The function $h : x \mapsto x \cdot \ln \left( x ^ { 2 } \right)$ is given with maximum domain $D _ { h }$. (1a) [2 marks] State $D _ { h }$ and show that for the term of the derivative function $h ^ { \prime }$ of $h$ the following holds: $h ^ { \prime } ( x ) = \ln \left( x ^ { 2 } \right) + 2$. (1b) [3 marks] Determine the coordinates of the local maximum point of the graph of $h$ located in the second quadrant. Figure 1 shows the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ of a polynomial function $f$ defined on $\mathbb { R }$. Only at the points $\left( - 4 \mid f ^ { \prime } ( - 4 ) \right)$ and $\left( 5 \mid f ^ { \prime } ( 5 ) \right)$ does the graph $G _ { f ^ { \prime } }$ have horizontal tangents. [Figure] (2a) [2 marks] Justify that $f$ has exactly one inflection point. (2b) [2 marks] There are tangents to the graph of $f$ that are parallel to the angle bisector of the first and third quadrants. Using the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ in Figure 1, determine approximate values for the $x$-coordinates of those points where the graph of $f$ has such a tangent. The functions $f : x \mapsto x ^ { 2 } + 4$ and $g _ { m } : x \mapsto m \cdot x$ with $m \in \mathbb { R }$ are given, both defined on $\mathbb { R }$. The graph of $f$ is denoted by $G _ { f }$ and the graph of $g _ { m }$ by $G _ { m }$. (3a) [3 marks] Sketch $G _ { f }$ in a coordinate system. Calculate the coordinates of the common point of the graphs $G _ { f }$ and $G _ { 4 }$. (3b) [2 marks] There are values of $m$ for which the graphs $G _ { f }$ and $G _ { m }$ have no common point. State these values of $m$. The function $g$ is given by $g ( x ) = 0,7 \cdot e ^ { 0,5 x } - 0,7$ with $x \in \mathbb { R }$. The function $g$ is invertible. Figure 2 shows the graph $G _ { g }$ of $g$ and part of the graph $G _ { h }$ of the inverse function $h$ of $g$. [Figure] (4a) [2 marks] Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$. Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$. The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]
The function $h : x \mapsto x \cdot \ln \left( x ^ { 2 } \right)$ is given with maximum domain $D _ { h }$.\\
\textbf{(1a)} [2 marks] State $D _ { h }$ and show that for the term of the derivative function $h ^ { \prime }$ of $h$ the following holds: $h ^ { \prime } ( x ) = \ln \left( x ^ { 2 } \right) + 2$.
\textbf{(1b)} [3 marks] Determine the coordinates of the local maximum point of the graph of $h$ located in the second quadrant.
Figure 1 shows the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ of a polynomial function $f$ defined on $\mathbb { R }$. Only at the points $\left( - 4 \mid f ^ { \prime } ( - 4 ) \right)$ and $\left( 5 \mid f ^ { \prime } ( 5 ) \right)$ does the graph $G _ { f ^ { \prime } }$ have horizontal tangents.\\
\textit{[Figure]}
\textbf{(2a)} [2 marks] Justify that $f$ has exactly one inflection point.
\textbf{(2b)} [2 marks] There are tangents to the graph of $f$ that are parallel to the angle bisector of the first and third quadrants. Using the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ in Figure 1, determine approximate values for the $x$-coordinates of those points where the graph of $f$ has such a tangent.
The functions $f : x \mapsto x ^ { 2 } + 4$ and $g _ { m } : x \mapsto m \cdot x$ with $m \in \mathbb { R }$ are given, both defined on $\mathbb { R }$. The graph of $f$ is denoted by $G _ { f }$ and the graph of $g _ { m }$ by $G _ { m }$.\\
\textbf{(3a)} [3 marks] Sketch $G _ { f }$ in a coordinate system. Calculate the coordinates of the common point of the graphs $G _ { f }$ and $G _ { 4 }$.
\textbf{(3b)} [2 marks] There are values of $m$ for which the graphs $G _ { f }$ and $G _ { m }$ have no common point. State these values of $m$.
The function $g$ is given by $g ( x ) = 0,7 \cdot e ^ { 0,5 x } - 0,7$ with $x \in \mathbb { R }$. The function $g$ is invertible. Figure 2 shows the graph $G _ { g }$ of $g$ and part of the graph $G _ { h }$ of the inverse function $h$ of $g$.\\
\textit{[Figure]}
\textbf{(4a)} [2 marks] Draw the missing part of $G _ { h }$ in Figure 2.\\
Sub-task Part A 4b $( 2 \mathrm { marks } )$\\
Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$.
Sub-task Part A 4c $( 2 \mathrm { marks } )$\\
State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$.
The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$.\\
\textit{[Figure]}