We propose to study the concentration in the blood of a medication ingested by a person for the first time. Let $t$ be the time (in hours) elapsed since the ingestion of this medication.\\
We admit that the concentration of this medication in the blood, in grams per litre of blood, is modelled by a function $f$ of the variable $t$ defined on the interval $[ 0 ; + \infty [$.

\section*{Part A: graphical readings}

The graph above shows the representative curve of the function $f$. With the precision allowed by the graph, give without justification:

\begin{enumerate}
  \item The time elapsed from the moment of ingestion of this medication to the moment when the concentration of medication in the blood is maximum according to this model.
  \item The set of solutions to the inequality $f ( t ) \geqslant 1$.
  \item The convexity of the function $f$ on the interval $[ 0 ; 8 ]$.
\end{enumerate}

\section*{Part B: determination of the function $\boldsymbol { f }$}
We consider the differential equation

$$( E ) : \quad y ^ { \prime } + y = 5 \mathrm { e } ^ { - t }$$

of unknown $y$, where $y$ is a function defined and differentiable on the interval $[ 0 ; + \infty [$. We admit that the function $f$ is a solution of the differential equation $( E )$.

\begin{enumerate}
  \item Solve the differential equation $\left( E ^ { \prime } \right) : y ^ { \prime } + y = 0$.
  \item Let $u$ be the function defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } u ( t ) = a t \mathrm { e } ^ { - t } \right. \right.$ with $a \in \mathbb { R }$.
\end{enumerate}

Determine the value of the real number $a$ such that the function $u$ is a solution of equation $( E )$.\\
3. Deduce the set of solutions of the differential equation $( E )$.\\
4. Since the person has not taken this medication before, we admit that $f ( 0 ) = 0$.

Determine the expression of the function $f$.

\section*{Part C: study of the function $\boldsymbol { f }$}
In this part, we admit that $f$ is defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$.

\begin{enumerate}
  \item Determine the limit of $f$ at $+ \infty$.
\end{enumerate}

Interpret this result in the context of the exercise.\\
2. Study the variations of $f$ on the interval $[ 0 ; + \infty [$ then draw up its complete variation table.\\
3. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 1$.

Give an approximate value to $10 ^ { - 2 }$ of the real numbers $t _ { 1 }$ and $t _ { 2 }$.\\
4. For a medication concentration greater than or equal to 1 gram per litre of blood, there is a risk of drowsiness.\\
What is the duration in hours and minutes of the drowsiness risk when taking this medication?

\section*{Part D: average concentration}
The average concentration of the medication (in grams per litre of blood) during the first hour is given by:

$$T _ { m } = \int _ { 0 } ^ { 1 } f ( t ) \mathrm { d } t$$

where $f$ is the function defined on $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$.\\
Calculate this average concentration.\\
Give the exact value then an approximate value to 0.01.