If $c$ is a point at which Rolle's theorem holds for the function, $f(\mathrm{x})=\log _{\mathrm{e}}\left(\frac{\mathrm{x}^{2}+\alpha}{7 \mathrm{x}}\right)$ in the interval $[3,4],$ where $\alpha \in \mathrm{R},$ then $f^{\prime \prime}(\mathrm{c})$ is equal to

If $g(x) = 2f (2x^3 - 3x^2) + f(6x^2 - 4x^3 - 3)$, $\forall  x \in R$ and $f"(x) > 0, \forall  x \in R$ , then $g'(x) > 0$ for $x$ belonging to

Let $\psi_1:[0, \infty) \rightarrow R , \psi_2:[0, \infty) \rightarrow R , f:[0, \infty) \rightarrow R$ and $g :[0, \infty) \rightarrow R$ be functions such that

$f(0)=g(0)=0$

$\Psi_1( x )= e ^{- x }+ x , \quad x \geq 0$

$\Psi_2( x )= x ^2-2 x -2 e ^{- x }+2, x \geq 0$

$f( x )=\int_{- x }^{ x }\left(| t |- t ^2\right) e ^{- t ^2} dt , x >0$

and

$g(x)=\int_0^{x^2} \sqrt{t} e^{-t} d t, x>0$

($1$) Which of the following statements is $TRUE$ ?

$(A)$ $f(\sqrt{\ln 3})+ g (\sqrt{\ln 3})=\frac{1}{3}$

$(B)$ For every $x>1$, there exists an $\alpha \in(1, x)$ such that $\psi_1(x)=1+\alpha x$

$(C)$ For every $x>0$, there exists a $\beta \in(0, x)$ such that $\psi_2(x)=2 x\left(\psi_1(\beta)-1\right)$

$(D)$ $f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$

($2$) Which of the following statements is $TRUE$ ?

$(A)$ $\psi_1$ (x) $\leq 1$, for all $x>0$

$(B)$ $\psi_2(x) \leq 0$, for all $x>0$

$(C)$ $f( x ) \geq 1- e ^{- x ^2}-\frac{2}{3} x ^3+\frac{2}{5} x ^5$, for all $x \in\left(0, \frac{1}{2}\right)$

$(D)$ $g(x) \leq \frac{2}{3} x^3-\frac{2}{5} x^5+\frac{1}{7} x^7$, for all $x \in\left(0, \frac{1}{2}\right)$

If the functions $f ( x )=\frac{ x ^3}{3}+2 bx +\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to

If $L.M.V.$ theorem is true for $f(x) = x(x-1)(x-2);\, x \in [0,\, 1/2]$ , then $C =$ ?

If the Rolle's theorem holds for the function $f(x) = 2x^3 + ax^2 + bx$ in the interval $[-1, 1 ]$ for the point $c = \frac{1}{2}$ , then the value of $2a + b$ is