For the function $f(x) = {e^x},a = 0,b = 1$, the value of $ c$ in mean value theorem will be

Suppose that $f (0) = - 3$ and $f ' (x) \le 5$ for all values of $x$. Then the largest value which $f (2)$ can attain is

Functions $f(x)$ and $g(x)$ are such that $f(x) + \int\limits_0^x {g(t)dt = 2\,\sin \,x\, - \,\frac{\pi }{2}} $ and $f'(x).g (x) = cos^2\,x$ , then number of solution $(s)$ of equation $f(x) + g(x) = 0$ in $(0,3 \pi$) is-

Let $f(x) = 8x^3 - 6x^2 - 2x + 1,$ then

Rolle's theorem is not applicable to the function $f(x) = |x|$ defined on $ [-1, 1] $ because

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