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 $j$.