\textbf{Exercise 1 (6 points) -- Part A}

Let $a$ and $b$ be real numbers. We consider a function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( x ) = \frac { a } { 1 + \mathrm { e } ^ { - b x } }$$
The curve $\mathscr { C } _ { f }$ representing the function $f$ in an orthogonal coordinate system is given. The curve $\mathscr { C } _ { f }$ passes through the point $\mathrm { A } ( 0 ; 0.5 )$. The tangent line to the curve $\mathscr { C } _ { f }$ at point A passes through the point B(10; 1).

\begin{enumerate}
  \item Justify that $a = 1$.
\end{enumerate}

We then obtain, for all real $x \geqslant 0$,
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - b x } }$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function. Verify that, for all real $x \geqslant 0$
$$f ^ { \prime } ( x ) = \frac { b \mathrm { e } ^ { - b x } } { \left( 1 + \mathrm { e } ^ { - b x } \right) ^ { 2 } }$$
  \item Using the data from the problem statement, determine $b$.
\end{enumerate}