Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by: $$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$ We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system. For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear. Is this the case?
Let $k$ be a strictly positive real number. Consider the functions $f _ { k }$ defined on $\mathbb { R }$ by:
$$f _ { k } ( x ) = x + k \mathrm { e } ^ { - x } .$$
We denote by $\mathscr { C } _ { k }$ the representative curve of function $f _ { k }$ in a plane with an orthonormal coordinate system.
For every strictly positive real number $k$, the function $f _ { k }$ admits a minimum on $\mathbb { R }$. The value at which this minimum is attained is the abscissa of the point denoted $A _ { k }$ on the curve $\mathscr { C } _ { k }$. It would seem that, for every strictly positive real number $k$, the points $A _ { k }$ are collinear.
Is this the case?