We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$. For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.
In this question, we choose $m = \mathrm{e}$. Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
Prove this conjecture.
We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$.
For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.
\begin{enumerate}
\item In this question, we choose $m = \mathrm{e}$.
Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
\item Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
\item Prove this conjecture.
\end{enumerate}