Throughout what follows, $m$ denotes any real number.
Part ALet $f$ be the function defined and differentiable on the set of real numbers $\mathbb{R}$ such that: $$f(x) = (x+1)\mathrm{e}^x$$
- Calculate the limit of $f$ at $+\infty$ and $-\infty$.
- We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$. Prove that for all real $x$, $f'(x) = (x+2)\mathrm{e}^x$.
- Draw the variation table of $f$ on $\mathbb{R}$.
Part BWe define the function $g_m$ on $\mathbb{R}$ by: $$g_m(x) = x + 1 - m\mathrm{e}^{-x}$$ and we denote by $\mathscr{C}_m$ the curve of function $g_m$ in a frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ of the plane.
- a. Prove that $g_m(x) = 0$ if and only if $f(x) = m$. b. Deduce from Part $A$, without justification, the number of intersection points of curve $\mathscr{C}_m$ with the $x$-axis as a function of the real number $m$.
- We have represented in appendix 2 the curves $\mathscr{C}_0$, $\mathscr{C}_{\mathrm{e}}$, and $\mathscr{C}_{-\mathrm{e}}$ (obtained by taking respectively for $m$ the values 0, e and $-$e). Identify each of these curves in the figure of appendix 2 by justifying.
- Study the position of curve $\mathscr{C}_m$ relative to the line $\mathscr{D}$ with equation $y = x + 1$ according to the values of the real number $m$.
- a. We call $D_2$ the part of the plane between curves $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $x = 2$. Shade $D_2$ on appendix 2. b. In this question, $a$ denotes a positive real number, $D_a$ the part of the plane between $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $\Delta_a$ with equation $x = a$. We denote by $\mathscr{A}(a)$ the area of this part of the plane, expressed in square units. Prove that for all positive real $a$: $\mathscr{A}(a) = 2\mathrm{e} - 2\mathrm{e}^{1-a}$. Deduce the limit of $\mathscr{A}(a)$ as $a$ tends to $+\infty$.