Composition du second semestre TL - 2024-2025
Exercice 1
1) On donne le polynôme $P(x)=ax^{3}+bx^{2}-18x+c$ ; où $a$,$b$ et $c$ sont des réels.
Déterminer $a$ ; $b$ et c sachant que $P\left(\dfrac{1}{2}\right)=0$, $(0)=8$ et $P(2)=0$
2. Dans la suite, on considère que $P(x)=2x^{3}+63x^{2}-18x+8$
a. Factoriser $P(x)=0$
b. Résoudre dans $\mathbb{R}$, l'équation $P(x)=0$
c. Résoudre dans $\mathbb{R}$, l'inéquation $p(x)\leq 0$
3. Déduire des questions précédentes les solutions dans $\mathbb{R}$ de :
a. $2\left(\ln(x+1)\right)^{3}+3\left(\ln(x+1)\right)^{2}-18\ln (x+1)+8=0$
b. $8\mathrm{e}^{-2x}-18\mathrm{e}^{-x}+2\mathrm{e}^{x}+3\leq 0$
Exercice 2
$\left(U_{n}\right)_{n\geq 0}$ est une suite arithmétique de raison $r$ telle que : $\left\lbrace\begin{array}{rcl} U_{8}+U_{9}+U_{10}&=&36\\ U_{11}&=&14 \end{array}\right.$
1. Montrer que $U_{0}=3$ et $r=1$
2. Exprimer $U_{n}$ en fonction de $n.$
En déduire la valeur de $U_{99}$
3. On pose $S_{n}=U_{0}+U_{1}+\ldots+U_{n}$
a. Exprimer $S_{n}$ en fonction de $n$
b. En déduire la valeur de $S=U_{0}+U_{1}+\ldots+U_{99}$
Exercice 3
On considère la fonction $f$ définie par : $f(x)=\dfrac{1+\ln x}{x}$
On désigne par $(C)$ sa courbe représentative dans le plan muni d'un repère orthonormé
1. Montrer que le domaine de définition de $f$ est $D_{f}=]0\ ;\ +\infty[$
2. Calculer les limites de $f$ aux bornes de son domaine de définition
3. Justifier que $(C)$ admet deux asymptotes dont on précisera, pour chacune d'elles, une équation.
4.a. Montrer que pour tout $x\in Df\;,f'(x)=\dfrac{-\ln x}{x^{2}}$, puis étudier son signe.
b. Dresser le tableau de variation de $f$
5. Tracer la courbe $(C)$ dans un repère orthonormé $\left(O\;,\vec{i}\;,\vec{j}\right)$
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