For each of the following statements, indicate whether it is true or false. You will justify each answer.

\textbf{Statement 1:} For all real numbers $a$ and $b$, $\left( \mathrm{e}^{a+b} \right)^{2} = \mathrm{e}^{2a} + \mathrm{e}^{2b}$.

\textbf{Statement 2:} In the plane with a coordinate system, the tangent line at point A with abscissa 0 to the representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = -2 + (3-x)\mathrm{e}^{x}$ has the reduced equation $y = 2x + 1$.

\textbf{Statement 3:} $\lim_{x \rightarrow +\infty} \left( \mathrm{e}^{2x} - \mathrm{e}^{x} + \frac{3}{x} \right) = 0$.

\textbf{Statement 4:} The equation $1 - x + \mathrm{e}^{-x} = 0$ has a unique solution belonging to the interval $[0 ; 2]$.

\textbf{Statement 5:} The function $g$ defined on $\mathbb{R}$ by $g(x) = x^{2} - 5x + \mathrm{e}^{x}$ is convex.