Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation
$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$
with unknown strictly positive real number $x$.

Part A

Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by
$$f ( x ) = \frac { \ln ( x ) } { x }$$
It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

1. Study the variations of function $f$.
2. Determine its maximum.

Part B

1. Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
2. From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$. a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$. Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ). b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$. Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$. c. Deduce that the sequence ( $\alpha _ { n }$ ) converges. It is not asked to calculate its limit.
3. It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that $$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$ a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing. Establish that, for every natural number $n$ greater than or equal to 3, $$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$ b. Deduce the limit of the sequence ( $\beta _ { n }$ ).

10. The graph of the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ is shown in the figure. Then the correct conclusion is [Figure]
(A) $a > 0 , b < 0 , c > 0 , d > 0$
(B) $a > 0 , b < 0 , c < 0 , d > 0$
(C) $a < 0 , b < 0 , c < 0 , d > 0$
(D) $a > 0 , b > 0 , c > 0 , d < 0$

II. Fill in the Blank Questions

(11) $\lg \frac { 5 } { 2 } + 2 \lg 2 - \left( \frac { 1 } { 2 } \right) ^ { - 1 } =$ $\_\_\_\_$. (12) In $\triangle A B C$, $A B = \sqrt { 6 } , \angle A = 75 ^ { \circ } , \angle B = 45 ^ { \circ }$. Then $A C =$ $\_\_\_\_$. (13) In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { n } = a _ { n - 1 } + \frac { 1 } { 2 } ( n \geq 2 )$. Then the sum of the first 9 terms of the sequence $\left\{ a _ { n } \right\}$ equals $\_\_\_\_$. (14) In the rectangular coordinate system $x O y$, if the line $y = 2 a$ and the graph of the function $y = | x - a | - 1$ have only one intersection point, then the value of $a$ is $\_\_\_\_$. (15) $\triangle A B C$ is an equilateral triangle with side length 2. Given that vectors $\vec { a } , \vec { b }$ satisfy $\overrightarrow { A B } = 2 \vec { a } , \overrightarrow { A C } = 2 \vec { a } + \vec { b }$, then the correct conclusions among the following are $\_\_\_\_$. (Write out the serial numbers of all correct conclusions)
(1) $\vec { a }$ is a unit vector; (2) $\vec { b }$ is a unit vector; (3) $\vec { a } \perp \vec { b }$; (4) $\vec { b } \parallel \overrightarrow { B C }$; (5) $( 4 \vec { a } + \vec { b } ) \perp \overrightarrow { B C }$.

III. Solution Questions

4. Which of the following functions is an increasing function?
A. $f ( x ) = - x$
B. $f ( x ) = \left( \frac { 2 } { 3 } \right) ^ { x }$
C. $f ( x ) = x ^ { 2 }$
D. $f ( x ) = \sqrt [ 3 ] { x }$

We are given four real numbers $a \leqslant b \leqslant c \leqslant d$ such that $a + d = b + c$. Study the variations of the function $x \mapsto |x - a| - |x - b| - |x - c| + |x - d|$; show that it takes positive values. A reasoned argument supported by a graphical representation would be welcome.

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1)\ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1) \ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0.$$
For all $n \in \mathbb { N } ^ { * }$, show that $B _ { n }$ is a decreasing function on $\mathbb { R } ^ { + }$. One may distinguish according to whether $n$ is even or odd.

The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is
(a) $( - \infty , 1 )$
(b) $( - \infty , - 1 )$
(c) $( 1 , \infty )$
(d) $( 3 , \infty )$

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only INCORRECT combination?
[A] (I) (iii) (P)
[B] (II) (iv) (Q)
[C] (III) (i) (R)
[D] (II) (iii) (P)

Let $f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$. Then the set of all values of $b$, for which $f(x)$ has maximum value at $x = 1$, is
(1) $(-2, -1]$
(2) $[-2, -1) \cup (1, 2]$
(3) $(-2, 2)$
(4) $(-\infty, -2) \cup (2, \infty)$