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

If from mean value theorem, $f'({x_1}) = {{f(b) – f(a)} \over {b – a}}$, then

Let $y = f (x)$ and $y = g (x)$ be two differentiable function in $[0,2]$ such that  $f(0) = 3,$ $f(2) = 5$ , $g (0) = 1$ and $g(2) = 2$. If there exist atlellst one $c \in \left( {0,2} \right)$ such that $f'(c)=kg'(c)$,then $k$ must be

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
  {{x^2}\ln x,\,x > 0} \\ 
  {0,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0} 
\end{array}} \right\}$, Rolle’s theorem is applicable to $ f $ for $x \in [0,1]$, if $\alpha = $