In this part, $k$ denotes a strictly positive real number. We consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = ( x - 1 ) \mathrm { e } ^ { - k x } + 1 .$$ We admit that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function. In the plane with an orthonormal coordinate system ( $\mathrm { O } ; \mathrm { I } , \mathrm { J }$ ), we denote $\mathscr { C } _ { f }$ the representative curve of the function $f$. The tangent line $T$ to the curve $\mathscr { C } _ { f }$ at point A with abscissa 1 intersects the ordinate axis at a point denoted B.
a. Prove that for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - k x } ( - k x + k + 1 )$$ b. Prove that the ordinate of point B is equal to $g ( k )$ where $g$ is the function defined in Part A, with $g ( x ) = 1 - \mathrm{e}^{-x}$.
Using Part A, prove that point B belongs to the segment [OJ].
\section*{Part B}
In this part, $k$ denotes a strictly positive real number.\\
We consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = ( x - 1 ) \mathrm { e } ^ { - k x } + 1 .$$
We admit that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function.\\
In the plane with an orthonormal coordinate system ( $\mathrm { O } ; \mathrm { I } , \mathrm { J }$ ), we denote $\mathscr { C } _ { f }$ the representative curve of the function $f$. The tangent line $T$ to the curve $\mathscr { C } _ { f }$ at point A with abscissa 1 intersects the ordinate axis at a point denoted B.
\begin{enumerate}
\item a. Prove that for all real $x$,
$$f ^ { \prime } ( x ) = \mathrm { e } ^ { - k x } ( - k x + k + 1 )$$
b. Prove that the ordinate of point B is equal to $g ( k )$ where $g$ is the function defined in Part A, with $g ( x ) = 1 - \mathrm{e}^{-x}$.
\item Using Part A, prove that point B belongs to the segment [OJ].
\end{enumerate}