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 $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Justify the existence of the real number $\Delta _ { n }$ for all $n \in \mathbb { N } ^ { * }$.

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.

Exercise III

Let $f$ be the function defined for every real number $x$ different from $1$ by $f ( x ) = \frac { 3 } { 1 - x }$ and $C _ { f }$ its representative curve in an orthonormal coordinate system. III-A- $\quad \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - \infty$. III-B- An equation of the tangent line to the curve $C _ { f }$ at the point with abscissa $x = - 1$ is $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$. III-C- $f$ is concave on $] 1 ; + \infty [$.
For each statement, indicate whether it is TRUE or FALSE.

Let $f \in \mathcal{C}(\mathbb{R})$ such that $$\lim_{x \rightarrow -\infty} f(x) = +\infty \quad \text{and} \quad \lim_{x \rightarrow +\infty} f(x) = +\infty$$ a) Show that the set $\{x \in \mathbb{R} \mid f(x) \leq f(0)\}$ is closed and bounded. b) Deduce that there exists $x_* \in \mathbb{R}$ such that $f(x_*) = \min\{f(x) \mid x \in \mathbb{R}\}$.

Draw the graph (on plain paper) of $f(x) = \min\{|x|-1, |x-1|-1, |x-2|-1\}$.

Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.

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) $( 3 , \infty )$

The equation $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y )$ represents
(a) Two parabolas intersecting at two points
(b) Two parabolas touching at one point
(c) Two non-intersecting hyperbolas
(d) One parabola passing through the origin.

Find the number of intersection points of $y = \log x$ and $y = x^2$.

Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true?
(A) $f$ has exactly two real roots
(B) $f$ has no real roots
(C) $f$ has four real roots
(D) $f$ has exactly two real roots, one of which is $-1$

How many real roots does $x ^ { 4 } + 12 x - 5$ have?

How many real roots does $x ^ { 4 } + 12 x - 5$ have?

For a real polynomial in one variable $P$, let $Z ( P )$ denote the locus of points ( $x , y$ ) in the plane such that $P ( x ) + P ( y ) = 0$. Then,
(A) there exist polynomials $Q _ { 1 }$ and $Q _ { 2 }$ such that $Z \left( Q _ { 1 } \right)$ is a circle and $Z \left( Q _ { 2 } \right)$ is a parabola.
(B) there does not exist any polynomial $Q$ such that $Z ( Q )$ is a circle or a parabola.
(C) there exists a polynomial $Q$ such that $Z ( Q )$ is a circle but there does not exist any polynomial $P$ such that $Z ( P )$ is a parabola.
(D) there exists a polynomial $Q$ such that $Z ( Q )$ is a parabola but there does not exist any polynomial $P$ such that $Z ( P )$ is a circle.

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

The number of real solutions of $e ^ { x } = \sin ( x )$ is
(A) 0
(B) 1
(C) 2
(D) infinite.

The number of real roots of the polynomial $$p ( x ) = \left( x ^ { 2020 } + 2020 x ^ { 2 } + 2020 \right) \left( x ^ { 3 } - 2020 \right) \left( x ^ { 2 } - 2020 \right)$$ is
(A) 2
(B) 3
(C) 2023
(D) 2025 .

For any real number $x$, let $[ x ]$ be the greatest integer $m$ such that $m \leq x$. Then the number of points of discontinuity of the function $g ( x ) = \left[ x ^ { 2 } - 2 \right]$ on the interval $( - 3,3 )$ is
(A) 5
(B) 9
(C) 13
(D) 16 .

The number of positive solutions to the equation $$e^x \sin x = \log x + e^{\sqrt{x}} + 2$$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

STATEMENT-1: The curve $y = \frac{-x^2}{2} + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2: A parabola is symmetric about its axis.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The real number $s$ lies in the interval
A) $\left( - \frac { 1 } { 4 } , 0 \right)$
B) $\left( - 11 , - \frac { 3 } { 4 } \right)$
C) $\left( - \frac { 3 } { 4 } , - \frac { 1 } { 2 } \right)$
D) $\left( 0 , \frac { 1 } { 4 } \right)$

If $$f ( x ) = \begin{cases} - x - \frac { \pi } { 2 } , & x \leq - \frac { \pi } { 2 } \\ - \cos x , & - \frac { \pi } { 2 } < x \leq 0 \\ x - 1 , & 0 < x \leq 1 \\ \ln x , & x > 1 , \end{cases}$$ then
(A) $f ( x )$ is continuous at $x = -\frac{\pi}{2}$
(B) $f ( x )$ is not differentiable at $x = 0$
(C) $f ( x )$ is differentiable at $x = 1$
(D) $f ( x )$ is differentiable at $x = -\frac{3}{2}$

The number of points in $( - \infty , \infty )$, for which $x ^ { 2 } - x \sin x - \cos x = 0$, is
(A) 6
(B) 4
(C) 2
(D) 0

For every pair of continuous functions $f, g : [0,1] \rightarrow \mathbb{R}$ such that $$\max\{f(x) : x \in [0,1]\} = \max\{g(x) : x \in [0,1]\}$$ the correct statement(s) is(are):
(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0,1]$
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0,1]$

Let $f : [0, 4\pi] \rightarrow [0, \pi]$ be defined by $f(x) = \cos^{-1}(\cos x)$. The number of points $x \in [0, 4\pi]$ satisfying the equation $$f(x) = \frac{10 - x}{10}$$ is