We consider the function $f$ defined on the interval $]-1; +\infty[$ by $$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$ We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.
Determine the limit of the function $f$ at $-1$.
Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have: $$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$. The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between). Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
Consider the following script, written in Python language: \begin{verbatim} from math import * def f(x): return 4*log(1+x)-(x**2)/25 def bornes(n) : p = 1/10**n x = 6 while f(x)-x > 0 : x = x + p return (x-p,x) \end{verbatim} We recall that in Python language:
the command $\log(x)$ returns the value $\ln x$;
the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth. b. Interpret these values in the context of the exercise.
Part B
In this part, we may use the results obtained in Part A. We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.
Show by induction that for all natural integer $n$, $$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$. Justify that $\ell = \alpha$.
\section*{Part A}
We consider the function $f$ defined on the interval $]-1; +\infty[$ by
$$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$
We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.
\begin{enumerate}
\item Determine the limit of the function $f$ at $-1$.
\item Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have:
$$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
\item Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
\item We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$.
The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between).
Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
\item Consider the following script, written in Python language:
\begin{verbatim}
from math import *
def f(x):
return 4*log(1+x)-(x**2)/25
def bornes(n) :
p = 1/10**n
x = 6
while f(x)-x > 0 :
x = x + p
return (x-p,x)
\end{verbatim}
We recall that in Python language:
\begin{itemize}
\item the command $\log(x)$ returns the value $\ln x$;
\item the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
\end{itemize}
a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth.\\
b. Interpret these values in the context of the exercise.
\end{enumerate}
\section*{Part B}
In this part, we may use the results obtained in Part A.\\
We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.
\begin{enumerate}
\item Show by induction that for all natural integer $n$,
$$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
\item Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
\item We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$.\\
Justify that $\ell = \alpha$.
\end{enumerate}