It is desired to create a gate. Each leaf measures 2 metres wide.

\section*{Part A: modelling the upper part of the gate}
The upper edge of the right leaf of the gate is modelled with a function $f$ defined on the interval [0;2] by

$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + b$$

where $b$ is a real number. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $[ 0 ; 2 ]$.

\begin{enumerate}
  \item a. Calculate $f ^ { \prime } ( x )$, for every real $x$ belonging to the interval $[ 0 ; 2 ]$.\\
b. Deduce the direction of variation of the function $f$ on the interval $[ 0 ; 2 ]$.
  \item Determine the number $b$ so that the maximum height of the gate is equal to $1{,}5 \mathrm{~m}$.
\end{enumerate}

In the following, the function $f$ is defined on the interval $[ 0 ; 2 ]$ by

$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 }$$

\section*{Part B: determination of an area}
Each leaf is made using a metal plate. We want to calculate the area of each plate, knowing that the lower edge of the leaf is at $0{,}05 \mathrm{~m}$ height from the ground.

\begin{enumerate}
  \item Show that the function $F$ defined on the interval $[ 0 ; 2 ]$ by
$$F ( x ) = \left( - \frac { 1 } { 4 } x - \frac { 5 } { 16 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 } x$$
is an antiderivative of $f$ on the interval $[ 0 ; 2 ]$.
  \item Calculate the area, in square metres, of each metal plate.
\end{enumerate}