Part A We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{6}{1 + 5e^{-x}}$$ We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.
Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have: $$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$ b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
We admit that:
$f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
for every real $x$,
$$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$ a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point. b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant. a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$. b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.
Part B The objective of this part is to study the following differential equation: $$(E) \quad y' = y - \frac{1}{6}y^2.$$ We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have: $$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$
Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
Solve the differential equation $y' = -y + \frac{1}{6}$.
We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish. We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$. We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$. a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$. b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by: $$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$ Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.
\textbf{Part A}\\
We consider the function $f$ defined on $\mathbb{R}$ by:
$$f(x) = \frac{6}{1 + 5e^{-x}}$$
We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.
\begin{enumerate}
\item Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
\item Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
\item a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have:
$$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$
b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
\item We admit that:
\begin{itemize}
\item $f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
\item for every real $x$,
\end{itemize}
$$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$
a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point.\\
b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
\item We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant.\\
a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$.\\
b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.
\end{enumerate}
\textbf{Part B}\\
The objective of this part is to study the following differential equation:
$$(E) \quad y' = y - \frac{1}{6}y^2.$$
We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have:
$$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$
\begin{enumerate}
\item Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
\item Solve the differential equation $y' = -y + \frac{1}{6}$.
\item We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish.\\
We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$.\\
We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$.\\
a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$.\\
b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by:
$$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$
Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.
\end{enumerate}