For each of the following statements, indicate whether it is true or false. Justify each answer.
Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$. Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system. Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$. Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.
For each of the following statements, indicate whether it is true or false. Justify each answer.
\begin{enumerate}
\item Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
\item We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$.\\
Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
\item We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system.\\
Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
\item We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$.\\
Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
\item Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
\item Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.
\end{enumerate}