Main topics covered: Exponential function; differentiation. The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by: $$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$ Question 3 is independent of questions 1 and 2.
a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$. b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$. We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function. a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$. b. Deduce the variations of the function $d$ on the interval $[-1; 1]$. c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
Let $\Delta$ be the line with equation $y = x + 2$. We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$. By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.
\textbf{Main topics covered: Exponential function; differentiation.}
The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$
\textbf{Question 3 is independent of questions 1 and 2.}
\begin{enumerate}
\item a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.\\
b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
\item For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$.\\
We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function.\\
a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$.\\
b. Deduce the variations of the function $d$ on the interval $[-1; 1]$.\\
c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
\item Let $\Delta$ be the line with equation $y = x + 2$.\\
We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$.\\
By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.
\end{enumerate}