\textbf{EXERCISE A}\\
Main topics covered: Exponential function, convexity, differentiation, differential equations

This exercise consists of three independent parts.\\
Below is represented, in an orthonormal coordinate system, a portion of the representative curve $\mathscr { C }$ of a function $f$ defined on $\mathbb { R }$.

Consider the points $\mathrm { A } ( 0 ; 2 )$ and $\mathrm { B } ( 2 ; 0 )$.

\section*{Part 1}
Knowing that the curve $\mathscr { C }$ passes through A and that the line (AB) is tangent to the curve $\mathscr { C }$ at point A, give by reading the graph:
\begin{enumerate}
  \item The value of $f ( 0 )$ and that of $f ^ { \prime } ( 0 )$.
  \item An interval on which the function $f$ appears to be convex.
\end{enumerate}

\section*{Part 2}
We denote $(E)$ the differential equation
$$y ^ { \prime } = - y + \mathrm { e } ^ { - x }$$
It is admitted that $g : x \longmapsto x \mathrm { e } ^ { - x }$ is a particular solution of $(E)$.
\begin{enumerate}
  \item Give all solutions on $\mathbb { R }$ of the differential equation $( H ) : y ^ { \prime } = - y$.
  \item Deduce all solutions on $\mathbb { R }$ of the differential equation $(E)$.
  \item Knowing that the function $f$ is the particular solution of $(E)$ which satisfies $f ( 0 ) = 2$, determine an expression of $f ( x )$ as a function of $x$.
\end{enumerate}

\section*{Part 3}
It is admitted that for every real number $x , f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }$.
\begin{enumerate}
  \item Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$.\\
a. Show that for all $x \in \mathbb { R } , f ^ { \prime } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$.\\
b. Study the sign of $f ^ { \prime } ( x )$ for all $x \in \mathbb { R }$ and draw up the table of variations of $f$ on $\mathbb { R }$.\\
Neither the limit of $f$ at $- \infty$ nor the limit of $f$ at $+ \infty$ will be specified.\\
Calculate the exact value of the extremum of $f$ on $\mathbb { R }$.
  \item Recall that $f ^ { \prime \prime }$ denotes the second derivative function of the function $f$.\\
a. Calculate for all $x \in \mathbb { R } , f ^ { \prime \prime } ( x )$.\\
b. Can we assert that $f$ is convex on the interval $[ 0 ; + \infty [$?
\end{enumerate}