Consider the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{-2x}$$ Let $f^{\prime\prime}$ denote the second derivative of the function $f$. For any real number $x$, $f^{\prime\prime}(x)$ is equal to: a. $(1-2x)\mathrm{e}^{-2x}$ b. $4(x-1)\mathrm{e}^{-2x}$ c. $4\mathrm{e}^{-2x}$ d. $(x+2)\mathrm{e}^{-2x}$

Let $f$ be the function defined for all real numbers $x$ in the interval $] 0 ; + \infty [$ by:
$$f ( x ) = \frac { \mathrm { e } ^ { 2 x } } { x }$$
The expression of the second derivative $f ^ { \prime \prime }$ of $f$ is given, defined on the interval $] 0 ; + \infty [$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } - 2 x + 1 \right) } { x ^ { 3 } } .$$

1. The function $f ^ { \prime }$, the derivative of $f$, is defined on the interval $] 0 ; + \infty [$ by: a. $f ^ { \prime } ( x ) = 2 \mathrm { e } ^ { 2 x }$ b. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( x - 1 ) } { x ^ { 2 } }$ c. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 2 x - 1 ) } { x ^ { 2 } }$ d. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 1 + 2 x ) } { x ^ { 2 } }$.
2. The function $f$: a. is decreasing on $] 0 ; + \infty [$ b. is monotonic on $] 0 ; + \infty [$ c. admits a minimum at $\frac { 1 } { 2 }$ d. admits a maximum at $\frac { 1 } { 2 }$.
3. The function $f$ has the following limit as $x \to + \infty$: a. $+ \infty$ b. 0 c. 1 d. $\mathrm { e } ^ { 2 x }$.
4. The function $f$: a. is concave on $] 0 ; + \infty [$ b. is convex on $] 0 ; + \infty [$ c. is concave on $] 0 ; \frac { 1 } { 2 } ]$ d. is represented by a curve admitting an inflection point.

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds: $$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$, $$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$

We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that $$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show that there exists a sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ with real coefficients such that $$\forall n \in \mathbb{N}, \forall x \in I, \quad f^{(n)}(x) = \frac{P_n(\sin x)}{(\cos x)^{n+1}}$$ Make explicit the polynomials $P_0, P_1, P_2, P_3$ and, for every natural integer $n$, express $P_{n+1}$ as a function of $P_n$ and $P_n^{\prime}$.

Let $v(x) = \frac{1}{\cos(x)}$ on $]-\frac{\pi}{2}, \frac{\pi}{2}[$ and $E_{2k} = v^{(2k)}(0)$ for $k \in \mathbb{N}$.
Show that, for $n \in \mathbb{N}^*$, $$\sum_{k=0}^{n} (-1)^k \binom{2n}{2k} E_{2k} = 0$$ and deduce the values of $E_0$, $E_2$ and $E_4$.

If $f ( x ) = e ^ { x } \sin x$, then $\left. \frac { d ^ { 10 } } { d x ^ { 10 } } f ( x ) \right| _ { x = 0 }$ equals
(a) 1 .
(B) - 1 .
(C) 10 .
(D) 32 .

Let $g ( x ) = \log f ( x )$ where $f ( x )$ is a twice differentiable positive function on $( 0 , \infty )$ such that $f ( x + 1 ) = x f ( x )$. Then, for $N = 1,2,3 , \ldots$,
$$g ^ { \prime \prime } \left( N + \frac { 1 } { 2 } \right) - g ^ { \prime \prime } \left( \frac { 1 } { 2 } \right) =$$
(A) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(B) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(C) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$
(D) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$

Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true?
A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$
B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$
C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$
D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$

List I P. Let $y(x) = \cos\left(3\cos^{-1}x\right)$, $x \in [-1,1]$, $x \neq \pm\frac{\sqrt{3}}{2}$. Then $\frac{1}{y(x)}\left\{\left(x^2-1\right)\frac{d^2y(x)}{dx^2} + x\frac{dy(x)}{dx}\right\}$ equals Q. Let $A_1, A_2, \ldots, A_n$ $(n > 2)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k$, $k = 1,2,\ldots,n$. If $\left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \times \overrightarrow{a_{k+1}}\right)\right| = \left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \cdot \overrightarrow{a_{k+1}}\right)\right|$, then the minimum value of $n$ is R. If the normal from the point $P(h,1)$ on the ellipse $\frac{x^2}{6} + \frac{y^2}{3} = 1$ is perpendicular to the line $x + y = 8$, then the value of $h$ is S. Number of positive solutions satisfying the equation $\tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{2}{x^2}\right)$ is
List II
1. 1
2. 2
3. 3
4. 4
P Q R S
(A) 4321
(B) 2431
(C) 4312
(D) 2413

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^5 \sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5 \cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$ The value of $\lambda$ for which $f''(0)$ exists is ____.
(A) 0
(B) 1
(C) $-1$
(D) $\frac{1}{2}$

If $y = \tan ^ { - 1 } \left( \sec x ^ { 3 } - \tan x ^ { 3 } \right) , \frac { \pi } { 2 } < x ^ { 3 } < \frac { 3 \pi } { 2 }$, then
(1) $x y ^ { \prime \prime } + 2 y ^ { \prime } = 0$
(2) $x ^ { 2 } y ^ { \prime \prime } - 6 y + \frac { 3 \pi } { 2 } = 0$
(3) $x ^ { 2 } y ^ { \prime \prime } - 6 y + 3 \pi = 0$
(4) $x y ^ { \prime \prime } - 4 y ^ { \prime } = 0$

If $y ( x ) = \left( x ^ { x } \right) ^ { x } , x > 0$ then $\frac { d ^ { 2 } x } { d y ^ { 2 } } + 20$ at $x = 1$ is equal to

Let $f ( x ) = \frac { \sin x + \cos x - \sqrt { 2 } } { \sin x - \cos x } , x \in [ 0 , \pi ] - \left\{ \frac { \pi } { 4 } \right\}$, then $f \left( \frac { 7 \pi } { 12 } \right) f ^ { \prime \prime } \left( \frac { 7 \pi } { 12 } \right)$ is equal to
(1) $\frac { 2 } { 9 }$
(2) $\frac { - 2 } { 3 }$
(3) $\frac { - 1 } { 3 \sqrt { 3 } }$
(4) $\frac { 2 } { 3 \sqrt { 3 } }$

Let $y = \log _ { e } \left( \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \right) , - 1 < x < 1$. Then at $x = \frac { 1 } { 2 }$, the value of $225 \left( y ^ { \prime } - y ^ { \prime \prime } \right)$ is equal to
(1) 732
(2) 746
(3) 742
(4) 736

If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$

If $y ( \theta ) = \frac { 2 \cos \theta + \cos 2 \theta } { \cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2 }$, then at $\theta = \frac { \pi } { 2 } , y ^ { \prime \prime } + y ^ { \prime } + y$ is equal to :
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 3 } { 2 }$

If $\log _ { e } y = 3 \sin ^ { - 1 } x$, then $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - x y ^ { \prime }$ at $x = \frac { 1 } { 2 }$ is equal to
(1) $3 e ^ { \pi / 6 }$
(2) $9 e ^ { \pi / 2 }$
(3) $3 e ^ { \pi / 2 }$
(4) $9 e ^ { \pi / 6 }$

$$f ( x ) = e ^ { 2 x } - e ^ { - 2 x }$$
What is the value of the 15th order derivative of the function at the point $x = \ln 2$, that is $\mathbf { f } ^ { \mathbf { ( 1 5 ) } } ( \mathbf { \ln } \mathbf { 2 } )$?
A) $17 \cdot 2 ^ { 13 }$
B) $15 \cdot 2 ^ { 13 }$
C) $9 \cdot 2 ^ { 13 }$
D) $15 \cdot 2 ^ { 12 }$
E) $7 \cdot 2 ^ { 12 }$

What is the value of the 16th order derivative $f ^ { ( 16 ) } ( x )$ of the function $f ( x ) = e ^ { x } \cdot \cos x$ at the point $x = 0$?
A) 32 B) 64 C) 128 D) 256 E) 512

Let $a$ and $b$ be real numbers. A function $f$ is defined on the set of positive real numbers as
$$f ( x ) = a x ^ { a } + b x ^ { b }$$
$$\begin{aligned} & f ( 1 ) = 6 \\ & f ^ { \prime } ( 1 ) = 20 \end{aligned}$$
Given that, what is $f''(1)$?
A) 44
B) 46
C) 48
D) 50
E) 52