\section*{Exercise 2 (6 points)}

We consider the function $f$ defined on $[ 0 ; + \infty [$ by

$$f ( x ) = 5 \mathrm { e } ^ { - x } - 3 \mathrm { e } ^ { - 2 x } + x - 3$$

We denote $\mathscr { C } _ { f }$ the graphical representation of the function $f$ and $\mathscr { D }$ the line with equation $y = x - 3$ in an orthogonal coordinate system of the plane.

\section*{Part A: Relative positions of $\mathscr { C } _ { f }$ and $\mathscr { D }$}
Let $g$ be the function defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = f ( x ) - ( x - 3 )$.

\begin{enumerate}
  \item Justify that, for every real number $x$ in the interval $[ 0 ; + \infty [ , g ( x ) > 0$.
  \item Do the curve $\mathscr { C } _ { f }$ and the line $\mathscr { D }$ have a common point? Justify.
\end{enumerate}

\section*{Part B: Study of the function $\boldsymbol { g }$}
We denote $M$ the point with abscissa $x$ of the curve $\mathscr { C } _ { f } , N$ the point with abscissa $x$ of the line $\mathscr { D }$ and we are interested in the evolution of the distance $M N$.

\begin{enumerate}
  \item Justify that, for all $x$ in the interval $[ 0 ; + \infty [$, the distance $M N$ is equal to $g ( x )$.
  \item We denote $g ^ { \prime }$ the derivative function of the function $g$ on the interval $[ 0 ; + \infty [$.
\end{enumerate}

For all $x$ in the interval $\left[ 0 ; + \infty \left[ \right. \right.$, calculate $g ^ { \prime } ( x )$.\\
3. Show that the function $g$ has a maximum on the interval $[ 0 ; + \infty [$ which we will determine.\\
Give a graphical interpretation of this.

\section*{Part C: Study of an area}
We consider the function $\mathscr { A }$ defined on the interval $[ 0 ; + \infty [$ by

$$\mathscr { A } ( x ) = \int _ { 0 } ^ { x } [ f ( t ) - ( t - 3 ) ] \mathrm { d } t$$

\begin{enumerate}
  \item Shade on the graph given in Appendix 1 (to be returned with your work) the region whose area is given by $\mathscr { A } ( 2 )$.
  \item Justify that the function $\mathscr { A }$ is increasing on the interval $[ 0 ; + \infty [$.
  \item For all positive real $x$, calculate $\mathscr { A } ( x )$.
  \item Does there exist a value of $x$ such that $\mathscr { A } ( x ) = 2$ ?
\end{enumerate}