We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right),$$
where $\ln$ denotes the natural logarithm function. We denote by $\mathscr{C}$ its representative curve in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

\begin{enumerate}
  \item a. Determine the limit of the function $f$ at $-\infty$.\\
b. Determine the limit of the function $f$ at $+\infty$.\\
Interpret this result graphically.\\
c. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. Calculate $f'(x)$ then show that, for every real number $x$, $f'(x) = \frac{-1}{1 + \mathrm{e}^x}$.\\
d. Draw the complete table of variations of the function $f$ on $\mathbb{R}$.
  \item We denote by $T_0$ the tangent line to the curve $\mathscr{C}$ at its point with abscissa 0.\\
a. Determine an equation of the tangent line $T_0$.\\
b. Show that the function $f$ is convex on $\mathbb{R}$.\\
c. Deduce that, for every real number $x$, we have:
$$f(x) \geqslant -\frac{1}{2}x + \ln(2)$$
  \item For every real number $a$ different from 0, we denote by $M_a$ and $N_a$ the points of the curve $\mathscr{C}$ with abscissas $-a$ and $a$ respectively.\\
We therefore have: $M_a(-a; f(-a))$ and $N_a(a; f(a))$.\\
a. Show that, for every real number $x$, we have: $f(x) - f(-x) = -x$.\\
b. Deduce that the lines $T_0$ and $(M_a N_a)$ are parallel.
\end{enumerate}