Let the function $f$ defined on the set of real numbers by $$f ( x ) = 2 \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$ and $\mathscr { C }$ its representative curve in an orthonormal coordinate system. We admit that, for all $x$ belonging to $[ 0 ; \ln ( 2 ) ] , f ( x )$ is positive. Indicate whether the following proposition is true or false by justifying your answer. Proposition A: The area of the region bounded by the lines with equations $x = 0$ and $x = \ln ( 2 )$, the $x$-axis and the curve $\mathscr { C }$ is equal to 1 unit of area.
Part B
Let $n$ be a strictly positive integer. Let the function $f _ { n }$ defined on the set of real numbers by $$f _ { n } ( x ) = 2 n \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$ and $\mathscr { C } _ { n }$ its representative curve in an orthonormal coordinate system. We admit that $f _ { n }$ is differentiable and that $\mathscr { C } _ { n }$ admits a horizontal tangent at a unique point $S _ { n }$. Indicate whether the following proposition is true or false by justifying your answer. Proposition B: For all strictly positive integer $n$, the ordinate of the point $S _ { n }$ is $n ^ { 2 }$.
The two parts of this exercise are independent.
\section*{Part A}
Let the function $f$ defined on the set of real numbers by
$$f ( x ) = 2 \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C }$ its representative curve in an orthonormal coordinate system.\\
We admit that, for all $x$ belonging to $[ 0 ; \ln ( 2 ) ] , f ( x )$ is positive.\\
Indicate whether the following proposition is true or false by justifying your answer.
\textbf{Proposition A:} The area of the region bounded by the lines with equations $x = 0$ and $x = \ln ( 2 )$, the $x$-axis and the curve $\mathscr { C }$ is equal to 1 unit of area.
\section*{Part B}
Let $n$ be a strictly positive integer.\\
Let the function $f _ { n }$ defined on the set of real numbers by
$$f _ { n } ( x ) = 2 n \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C } _ { n }$ its representative curve in an orthonormal coordinate system.\\
We admit that $f _ { n }$ is differentiable and that $\mathscr { C } _ { n }$ admits a horizontal tangent at a unique point $S _ { n }$.\\
Indicate whether the following proposition is true or false by justifying your answer.
\textbf{Proposition B:} For all strictly positive integer $n$, the ordinate of the point $S _ { n }$ is $n ^ { 2 }$.