Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$
We denote by $P_n^{(n)}$ the polynomial derived $n$ times of $P_n$.
Determine the degree of $P_n^{(n)}$ and calculate $P_n^{(n)}(1)$.

Show the equality: for all $m \geq 0$ and $\mu \geq 0$, $$\left(\frac{d}{dx}\right)^{\mu} \cdot \left(x^m f\right) = \left(x^m \left(\frac{d}{dx}\right)^{\mu} + \sum_{i=1}^{\min(\mu,m)} \frac{m(m-1)\cdots(m-i+1)\,\mu(\mu-1)\cdots(\mu-i+1)}{i!} x^{m-i} \left(\frac{d}{dx}\right)^{\mu-i}\right) \cdot f.$$

48. $\frac { d ^ { 2 } x } { d y ^ { 2 } }$ equals
(A) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 }$
(B) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 } \left( \frac { d y } { d x } \right) ^ { - 3 }$
(C) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 2 }$
(D) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 3 }$

Answer

◯ ◯
(A)
(B)
(C)
(D)

Let $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$.
STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$. and STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$.
(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

If $y = e ^ { n x }$, then $\frac { d ^ { 2 } y } { d x ^ { 2 } } \cdot \frac { d ^ { 2 } x } { d y ^ { 2 } }$ is equal to:
(1) $n e ^ { - n x }$
(2) $- n e ^ { - n x }$
(3) $n e ^ { n x }$
(4) 1

Let, $f : R \rightarrow R$ be a function such that $f ( x ) = x ^ { 3 } + x ^ { 2 } f \prime ( 1 ) + x f \prime \prime ( 2 ) + f \prime \prime \prime ( 3 ) , \forall x \in R$. Then $f ( 2 )$ equals
(1) 30
(2) 8
(3) $- 4$
(4) $- 2$

If $f(x) = x^2 + g'(1)x + g''(2)$ and $g(x) = f(1)x^2 + xf'(x) + f''(x)$, then the value of $f(4) - g(4)$ is equal to $\_\_\_\_$.

Let $y ( x ) = ( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 8 } \right) \left( 1 + x ^ { 16 } \right)$. Then $y ^ { \prime } - y ^ { \prime \prime }$ at $x = - 1$ is equal to
(1) 976
(2) 464
(3) 496
(4) 944

Let $f ( x ) = x ^ { 3 } + x ^ { 2 } f ^ { \prime } ( 1 ) + x f ^ { \prime \prime } ( 2 ) + f ^ { \prime \prime \prime } ( 3 ) , x \in R$. Then $f ^ { \prime } ( 10 )$ is equal to