In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we place ourselves on the time interval $[0;10]$.

\section*{Part A}
In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[0; 10]$.
\begin{enumerate}
  \item Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$.
\end{enumerate}
We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[0; 10]$. We admit that the function $f$ can be written in the form $f(t) = (at + b)\mathrm{e}^{-0.5t}$ where $a$ and $b$ are two real constants.\\
2. We admit that the exact value of $f(0)$ is 40. Deduce the value of $b$.\\
3. We admit that $f$ satisfies the differential equation (E): $y' + 0.5y = 60\mathrm{e}^{-0.5t}$. Determine the value of $a$.

\section*{Part B: Study of the function $f$}
We admit that the function $f$ is defined for every real $t$ in the interval $[0; 10]$ by
$$f(t) = (60t + 40)\mathrm{e}^{-0.5t}$$
\begin{enumerate}
  \item Show that for every real $t$ in the interval $[0; 10]$, we have: $f'(t) = (40 - 30t)\mathrm{e}^{-0.5t}$.
  \item a. Study the direction of variation of the function $f$ on the interval $[0; 10]$. Draw the variation table of the function $f$ showing the images of the values present in the table.\\
b. Show that the equation $f(t) = 40$ has a unique solution $\alpha$ strictly positive on the interval $]0; 10]$.\\
c. Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation in the context of the exercise.
  \item We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t_{1}$ and $t_{2}$, expressed in minutes, by
$$\frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} f(t)\,\mathrm{dt}$$
a. Using integration by parts, show that
$$\int_{0}^{4} f(t)\,\mathrm{dt} = 320 - \frac{800}{\mathrm{e}^{2}}$$
b. Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.
\end{enumerate}