(1) Determine the coordinates of the inflection point of the graph of $f _ { k }$ as a function of $k$ by calculation. [For verification: For the $x$-coordinate $x _ { w }$ of the inflection point, we have: $x _ { w } = k$.] The inflection points of the graphs of $f _ { k }$ with $k \geq - 0,5$ lie on the graph of the function $w$ with $w ( x ) = \mathrm { e } ^ { - x } \cdot x ^ { 2 } , x \geq - 0,5$. The graph of $w$ is called the locus curve of the inflection points of the function family. (2) (i) Show: $w ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$. (ii) Determine the global maximum of $w$ by calculation. (3) Given is the function $w _ { \text {new } }$ with the equation
$$w _ { \text {new } } ( x ) = 3 \cdot \mathrm { e } ^ { - ( x - 2 ) } \cdot ( x - 2 ) ^ { 2 } , x \in \mathbb { R } .$$
The graph of $w _ { \text {new } }$ is the locus curve of the inflection points of another family of functions $v _ { k }$ with $k \geq - 0,5$.
Give a possible function equation for $v _ { k }$.