If $f(x) = \cos x,0 \le x \le {\pi \over 2}$, then the real number $ ‘c’ $ of the mean value theorem is

If $f(x)$ satisfies the conditions of Rolle’s theorem in $[1,\,2]$ and $f(x)$ is continuous in $[1,\,2]$ then $\int_1^2 {f'(x)dx} $ is equal to

Let $f: R \rightarrow R$ be a differentiable function such that $f(a)=0=f(b)$ and $f^{\prime}(a) f^{\prime}(b) > 0$ for some $a < b$. Then, the minimum number of roots of $f^{\prime}(x)=0$ in the interval $(a, b)$ is

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 $x \in\left(0, \frac{1}{2}\right)$

Let $g: R \rightarrow R$ be a non constant twice differentiable such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $\mathrm{f}(\mathrm{x})=\frac{1}{2}[\mathrm{~g}(\mathrm{x})+\mathrm{g}(2-\mathrm{x})]$, then

In  $[0, 1]$ Lagrange's mean value theorem is $ NOT$  applicable to