Let $f$ be a function defined and differentiable on $\mathbb { R }$. We consider the points $\mathrm { A } ( 1 ; 3 )$ and $\mathrm { B } ( 3 ; 5 )$. We give below $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthogonal coordinate system of the plane, as well as the tangent line (AB) to the curve $\mathscr { C } _ { f }$ at point A.

The three parts of the exercise can be worked on independently.

\section*{Part A}
\begin{enumerate}
  \item Determine graphically the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
  \item The function $f$ is defined by the expression $f ( x ) = \ln \left( a x ^ { 2 } + 1 \right) + b$, where $a$ and $b$ are positive real numbers.\\
a. Determine the expression of $f ^ { \prime } ( x )$.\\
b. Determine the values of $a$ and $b$ using the previous results.
\end{enumerate}

\section*{Part B}
It is admitted that the function $f$ is defined on $\mathbb { R }$ by
$$f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$$
\begin{enumerate}
  \item Show that $f$ is an even function.
  \item Determine the limits of $f$ at $+ \infty$ and at $- \infty$.
  \item Determine the expression of $f ^ { \prime } ( x )$. Study the direction of variation of the function $f$ on $\mathbb { R }$.\\
Draw up the table of variations of $f$ showing the exact value of the minimum as well as the limits of $f$ at $- \infty$ and $+ \infty$.
  \item Using the table of variations of $f$, give the values of the real number $k$ for which the equation $f ( x ) = k$ admits two solutions.
  \item Solve the equation $f ( x ) = 3 + \ln 2$.
\end{enumerate}

\section*{Part C}
We recall that the function $f$ is defined on $\mathbb{R}$ by $f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$.
\begin{enumerate}
  \item Conjecture, by graphical reading, the abscissas of any inflection points of the curve $\mathscr { C } _ { f }$.
  \item Show that, for any real number $x$, we have: $f ^ { \prime \prime } ( x ) = \frac { 2 \left( 1 - x ^ { 2 } \right) } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$.
  \item Deduce the largest interval on which the function $f$ is convex.
\end{enumerate}