Let $f$ be the function defined on the interval $[0; +\infty[$ by $$f(x) = x\mathrm{e}^{-x}$$ We denote by $\mathscr{C}$ the representative curve of $f$ in an orthogonal coordinate system.
Part A
We denote by $f'$ the derivative function of the function $f$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $f'(x)$. Deduce the variations of the function $f$ on the interval $[0; +\infty[$.
Determine the limit of the function $f$ at $+\infty$. What graphical interpretation can be made of this result?
Part B
Let $\mathscr{A}$ be the function defined on the interval $[0; +\infty[$ as follows: for every real number $t$ in the interval $[0; +\infty[$, $\mathscr{A}(t)$ is the area, in square units, of the region bounded by the $x$-axis, the curve $\mathscr{C}$ and the lines with equations $x = 0$ and $x = t$.
Determine the direction of variation of the function $\mathscr{A}$.
We admit that the area of the region bounded by the curve $\mathscr{C}$ and the $x$-axis is equal to 1 square unit. What can we deduce about the function $\mathscr{A}$?
We seek to prove the existence of a real number $\alpha$ such that the line with equation $x = \alpha$ divides the region between the $x$-axis and the curve $\mathscr{C}$ into two parts of equal area, and to find an approximate value of this real number. a) Prove that the equation $\mathscr{A}(t) = \frac{1}{2}$ has a unique solution on the interval $[0; +\infty[$ b) On the graph provided in the appendix (to be returned with the answer sheet) are drawn the curve $\mathscr{C}$, as well as the curve $\Gamma$ representing the function $\mathscr{A}$. On the graph in the appendix, identify the curves $\mathscr{C}$ and $\Gamma$, then draw the line with equation $y = \frac{1}{2}$. Deduce an approximate value of the real number $\alpha$. Shade the region corresponding to $\mathscr{A}(\alpha)$.
We define the function $g$ on the interval $[0; +\infty[$ by $$g(x) = (x + 1)\mathrm{e}^{-x}$$ a) We denote by $g'$ the derivative function of the function $g$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $g'(x)$. b) Deduce, for every real number $t$ in the interval $[0; +\infty[$, an expression for $\mathscr{A}(t)$. c) Calculate an approximate value to $10^{-2}$ of $\mathscr{A}(6)$.
Let $f$ be the function defined on the interval $[0; +\infty[$ by
$$f(x) = x\mathrm{e}^{-x}$$
We denote by $\mathscr{C}$ the representative curve of $f$ in an orthogonal coordinate system.
\section*{Part A}
\begin{enumerate}
\item We denote by $f'$ the derivative function of the function $f$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $f'(x)$. Deduce the variations of the function $f$ on the interval $[0; +\infty[$.
\item Determine the limit of the function $f$ at $+\infty$. What graphical interpretation can be made of this result?
\end{enumerate}
\section*{Part B}
Let $\mathscr{A}$ be the function defined on the interval $[0; +\infty[$ as follows: for every real number $t$ in the interval $[0; +\infty[$, $\mathscr{A}(t)$ is the area, in square units, of the region bounded by the $x$-axis, the curve $\mathscr{C}$ and the lines with equations $x = 0$ and $x = t$.
\begin{enumerate}
\item Determine the direction of variation of the function $\mathscr{A}$.
\item We admit that the area of the region bounded by the curve $\mathscr{C}$ and the $x$-axis is equal to 1 square unit. What can we deduce about the function $\mathscr{A}$?
\item We seek to prove the existence of a real number $\alpha$ such that the line with equation $x = \alpha$ divides the region between the $x$-axis and the curve $\mathscr{C}$ into two parts of equal area, and to find an approximate value of this real number.\\
a) Prove that the equation $\mathscr{A}(t) = \frac{1}{2}$ has a unique solution on the interval $[0; +\infty[$\\
b) On the graph provided in the appendix (to be returned with the answer sheet) are drawn the curve $\mathscr{C}$, as well as the curve $\Gamma$ representing the function $\mathscr{A}$. On the graph in the appendix, identify the curves $\mathscr{C}$ and $\Gamma$, then draw the line with equation $y = \frac{1}{2}$. Deduce an approximate value of the real number $\alpha$. Shade the region corresponding to $\mathscr{A}(\alpha)$.
\item We define the function $g$ on the interval $[0; +\infty[$ by
$$g(x) = (x + 1)\mathrm{e}^{-x}$$
a) We denote by $g'$ the derivative function of the function $g$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $g'(x)$.\\
b) Deduce, for every real number $t$ in the interval $[0; +\infty[$, an expression for $\mathscr{A}(t)$.\\
c) Calculate an approximate value to $10^{-2}$ of $\mathscr{A}(6)$.
\end{enumerate}