Let $f$ be the function defined on the interval $]0;+\infty[$ by: $$f(x) = \frac{\mathrm{e}^{x}}{x}.$$ We denote $\mathscr{C}_{f}$ the representative curve of the function $f$ in an orthonormal coordinate system.
a. Specify the limit of the function $f$ at $+\infty$. b. Justify that the $y$-axis is an asymptote to the curve $\mathscr{C}_{f}$.
Show that, for every real number $x$ in the interval $]0;+\infty[$, we have: $$f^{\prime}(x) = \frac{\mathrm{e}^{x}(x-1)}{x^{2}}$$ where $f^{\prime}$ denotes the derivative function of the function $f$.
Determine the variations of the function $f$ on the interval $]0;+\infty[$. A variation table of the function $f$ will be established in which the limits appear.
Let $m$ be a real number. Specify, depending on the values of the real number $m$, the number of solutions of the equation $f(x) = m$.
We denote $\Delta$ the line with equation $y = -x$. We denote A a possible point of $\mathscr{C}_{f}$ with abscissa $a$ at which the tangent to the curve $\mathscr{C}_{f}$ is parallel to the line $\Delta$. a. Show that $a$ is a solution of the equation $\mathrm{e}^{x}(x-1) + x^{2} = 0$. We denote $g$ the function defined on $[0;+\infty[$ by $g(x) = \mathrm{e}^{x}(x-1) + x^{2}$. We assume that the function $g$ is differentiable and we denote $g^{\prime}$ its derivative function. b. Calculate $g^{\prime}(x)$ for every real number $x$ in the interval $[0;+\infty[$, then establish the variation table of $g$ on $[0;+\infty[$. c. Show that there exists a unique point $A$ at which the tangent to $\mathscr{C}_{f}$ is parallel to the line $\Delta$.
Let $f$ be the function defined on the interval $]0;+\infty[$ by:
$$f(x) = \frac{\mathrm{e}^{x}}{x}.$$
We denote $\mathscr{C}_{f}$ the representative curve of the function $f$ in an orthonormal coordinate system.
\begin{enumerate}
\item a. Specify the limit of the function $f$ at $+\infty$.\\
b. Justify that the $y$-axis is an asymptote to the curve $\mathscr{C}_{f}$.
\item Show that, for every real number $x$ in the interval $]0;+\infty[$, we have:
$$f^{\prime}(x) = \frac{\mathrm{e}^{x}(x-1)}{x^{2}}$$
where $f^{\prime}$ denotes the derivative function of the function $f$.
\item Determine the variations of the function $f$ on the interval $]0;+\infty[$. A variation table of the function $f$ will be established in which the limits appear.
\item Let $m$ be a real number. Specify, depending on the values of the real number $m$, the number of solutions of the equation $f(x) = m$.
\item We denote $\Delta$ the line with equation $y = -x$.
We denote A a possible point of $\mathscr{C}_{f}$ with abscissa $a$ at which the tangent to the curve $\mathscr{C}_{f}$ is parallel to the line $\Delta$.\\
a. Show that $a$ is a solution of the equation $\mathrm{e}^{x}(x-1) + x^{2} = 0$.
We denote $g$ the function defined on $[0;+\infty[$ by $g(x) = \mathrm{e}^{x}(x-1) + x^{2}$.\\
We assume that the function $g$ is differentiable and we denote $g^{\prime}$ its derivative function.\\
b. Calculate $g^{\prime}(x)$ for every real number $x$ in the interval $[0;+\infty[$, then establish the variation table of $g$ on $[0;+\infty[$.\\
c. Show that there exists a unique point $A$ at which the tangent to $\mathscr{C}_{f}$ is parallel to the line $\Delta$.
\end{enumerate}