A function $f$ is given by the function equation $f ( x ) = x ^ { 3 } - 3 x , x \in \mathbb { R }$. The graph of $f$ is shown in the figure. (1) Justify that the graph of $f$ is point-symmetric about the origin. (2) The graph of $f$ encloses an area with the $x$-axis in the second quadrant. Determine the area $A$ of this region by calculation. [For verification: $A = 2.25 \mathrm { FE }$.] (3) Given is the line $g : y = - 2 x , x \in \mathbb { R }$. Determine the ratio in which the line $g$ divides the area from (2).
For each $k \in \mathbb { R }$, a polynomial function defined on $\mathbb { R }$ is given by $$u ( x ) = x ^ { 3 } - 3 k \cdot x + k ^ { 2 } - 1$$ (1) Give the value of $k$ for which the corresponding function $u$ coincides with the function $f$. (2) Determine all values of $k$ for which $u ( 2 ) = 2$ holds.
Given is the function $h$ with $h ( x ) = x ^ { 2 } \cdot \mathrm { e } ^ { - x } , x \in \mathbb { R }$. (1) (i) Show: $h ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$. (ii) Calculate the coordinates and the type of local extreme points of the graph of $h$. [For verification: The local maximum point of $h$ is $x = 2$.] (2) The points $P ( 0 \mid 0 ) , Q ( r \mid 0 )$ and $R ( r \mid h ( r ) )$ form the vertices of a triangle $P Q R$ for $0 \leq r \leq 10$. Determine $r$ so that the area of triangle $P Q R$ is maximal. (3) Describe how the graph of $j$ with $j ( x ) = 3 \cdot ( x - 2 ) ^ { 2 } \cdot \mathrm { e } ^ { - ( x - 2 ) }$ is obtained from the graph of $h$. Give the local maximum point of the graph of the function $j$.