A certification body is commissioned to evaluate two heating devices, one from brand A and the other from brand B.

\textbf{Parts 1 and 2 are independent.}

\textbf{Part 1: device from brand A}\\
Using a probe, the temperature inside the combustion chamber of a brand A device was measured. Below is a representation of the temperature curve in degrees Celsius inside the combustion chamber as a function of time elapsed, expressed in minutes, since the combustion chamber was ignited.

By reading the graph:
\begin{enumerate}
  \item Give the time at which the maximum temperature is reached inside the combustion chamber.
  \item Give an approximate value, in minutes, of the duration during which the temperature inside the combustion chamber exceeds $300 ^ { \circ } \mathrm { C }$.
  \item We denote by $f$ the function represented on the graph. Estimate the value of $\frac { 1 } { 600 } \int _ { 0 } ^ { 600 } f ( t ) \mathrm { d } t$. Interpret the result.
\end{enumerate}

\textbf{Part 2: study of a function}\\
Let the function $g$ be defined on the interval $[0 ; + \infty [$ by:
$$g ( t ) = 10 t \mathrm { e } ^ { - 0.01 t } + 20 .$$
\begin{enumerate}
  \item Determine the limit of $g$ at $+ \infty$.
  \item a. Show that for all $t \in \left[ 0 ; + \infty \left[ , \quad g ^ { \prime } ( t ) = ( - 0.1 t + 10 ) \mathrm { e } ^ { - 0.01 t } \right. \right.$.\\
b. Study the variations of the function $g$ on $[0 ; + \infty [$ and construct its variation table.
  \item Prove that the equation $g ( t ) = 300$ has exactly two distinct solutions on $[0 ; + \infty [$. Give approximate values to the nearest integer.
  \item Using integration by parts, calculate $\int _ { 0 } ^ { 600 } g ( t ) \mathrm { d } t$.
\end{enumerate}

\textbf{Part 3: evaluation}\\
For a brand B device, the temperature in degrees Celsius inside the combustion chamber $t$ minutes after ignition is modelled on $[0 ; 600]$ by the function $g$.

The certification body awards one star for each criterion validated among the following four:
\begin{itemize}
  \item Criterion 1: the maximum temperature is greater than $320 ^ { \circ } \mathrm { C }$.
  \item Criterion 2: the maximum temperature is reached in less than 2 hours.
  \item Criterion 3: the average temperature during the first 10 hours after ignition exceeds $250 ^ { \circ } \mathrm { C }$.
  \item Criterion 4: the temperature inside the combustion chamber must not exceed $300 ^ { \circ } \mathrm { C }$ for more than 5 hours.
\end{itemize}
Does each device obtain exactly three stars? Justify your answer.