The function $f : x \mapsto 8 x ^ { 3 } + 3 x$ is defined on $\mathbb { R }$ with derivative function $f ^ { \prime }$. (1a) [2 marks] Calculate $f ^ { \prime } ( 1 )$. (1b) [3 marks] Determine a term for the antiderivative $F$ of $f$ whose graph passes through the point ( $- 1 \mid 5$ ). The figure shows the graph $G _ { g }$ of the function $g$ defined on $\mathbb { R }$ with $g ( x ) = 2 \cdot \sin \left( \frac { 1 } { 2 } x \right)$. [Figure] (2a) [2 marks] Using the figure, assess whether the value of the integral $\int _ { - 2 } ^ { 8 } g ( x ) \mathrm { dx }$ is negative. (2b) [3 marks] Prove by calculation that the following statement is true: The tangent to $G _ { g }$ at the origin is the line through the points $( - 1 \mid - 1 )$ and $( 1 \mid 1 )$. Consider the family of functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = x \cdot e ^ { a x }$ and $a \in \mathbb { R } \backslash \{ 0 \}$. For each value of $a$, the function $f _ { a }$ has exactly one extremum. (3a) [2 marks] Justify that the graph of $f _ { a }$ lies below the x-axis for $x < 0$. (3b) [3 marks] The displayed graphs I and II are graphs of the family; one of the two belongs to a positive value of $a$. Decide which graph this is and justify your decision. [Figure] (4a) [2 marks] Give a term for a function $g$ defined on $\mathbb { R }$ that has range $[ - 2 ; 4 ]$. (4b) [3 marks] Give a term for a function $h$ defined on $\mathbb { R }$ such that the term $\sqrt { h ( x ) }$ is defined exactly for $x \in [ - 2 ; 4 ]$. Explain the reasoning underlying your answer. The figure shows a section of the graph $G$ of the function $f$ defined on $\mathbb { R } \backslash \{ - 3 \}$ with $f ( x ) = x - 3 + \frac { 5 } { x + 3 }$. $G$ has exactly one minimum point $T$. [Figure]
The function $f : x \mapsto 8 x ^ { 3 } + 3 x$ is defined on $\mathbb { R }$ with derivative function $f ^ { \prime }$.\\
\textbf{(1a)} [2 marks] Calculate $f ^ { \prime } ( 1 )$.
\textbf{(1b)} [3 marks] Determine a term for the antiderivative $F$ of $f$ whose graph passes through the point ( $- 1 \mid 5$ ).
The figure shows the graph $G _ { g }$ of the function $g$ defined on $\mathbb { R }$ with $g ( x ) = 2 \cdot \sin \left( \frac { 1 } { 2 } x \right)$.\\
\textit{[Figure]}
\textbf{(2a)} [2 marks] Using the figure, assess whether the value of the integral $\int _ { - 2 } ^ { 8 } g ( x ) \mathrm { dx }$ is negative.
\textbf{(2b)} [3 marks] Prove by calculation that the following statement is true:\\
The tangent to $G _ { g }$ at the origin is the line through the points $( - 1 \mid - 1 )$ and $( 1 \mid 1 )$.
Consider the family of functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = x \cdot e ^ { a x }$ and $a \in \mathbb { R } \backslash \{ 0 \}$. For each value of $a$, the function $f _ { a }$ has exactly one extremum.
\textbf{(3a)} [2 marks] Justify that the graph of $f _ { a }$ lies below the x-axis for $x < 0$.\\
\textbf{(3b)} [3 marks] The displayed graphs I and II are graphs of the family; one of the two belongs to a positive value of $a$. Decide which graph this is and justify your decision.\\
\textit{[Figure]}
\textbf{(4a)} [2 marks] Give a term for a function $g$ defined on $\mathbb { R }$ that has range $[ - 2 ; 4 ]$.
\textbf{(4b)} [3 marks] Give a term for a function $h$ defined on $\mathbb { R }$ such that the term $\sqrt { h ( x ) }$ is defined exactly for $x \in [ - 2 ; 4 ]$. Explain the reasoning underlying your answer.
The figure shows a section of the graph $G$ of the function $f$ defined on $\mathbb { R } \backslash \{ - 3 \}$ with $f ( x ) = x - 3 + \frac { 5 } { x + 3 }$. $G$ has exactly one minimum point $T$.\\
\textit{[Figure]}