If the function $f(x) = x^3 - 6x^2 + ax + b$ satisfies Rolle's theorem in the interval $[1, 3]$ and $f'\left( \frac{2\sqrt{3} + 1}{\sqrt{3}} \right) = 0$,then $a = $ ..............

If the function $f(x)=\sqrt{x^2-4}$ satisfies the Lagrange's mean value theorem on $[2, 4]$,then the value of $C$ is

Let $f$ be a function which is continuous and differentiable for all $x$. If $f(1) = 1$ and $f^{\prime}(x) \leq 5$ for all $x$ in $[1, 5]$,then the maximum value of $f(5)$ is

The value of $c$ for the Lagrange's mean value theorem for $f(x)=\sqrt{x^2-x}, x \in[1,4]$ is

Examine the applicability of the Mean Value Theorem for the following functions:
$(i)$ $f(x) = [x]$ for $x \in [5, 9]$
$(ii)$ $f(x) = [x]$ for $x \in [-2, 2]$
$(iii)$ $f(x) = x^{2} - 1$ for $x \in [1, 2]$

Let $\psi_1:[0, \infty) \rightarrow \mathbb{R}$,$\psi_2:[0, \infty) \rightarrow \mathbb{R}$,$f:[0, \infty) \rightarrow \mathbb{R}$,and $g:[0, \infty) \rightarrow \mathbb{R}$ be functions such that $f(0)=g(0)=0$,$\psi_1(x)=e^{-x}+x$ for $x \geq 0$,$\psi_2(x)=x^2-2x-2e^{-x}+2$ for $x \geq 0$,$f(x)=\int_{-x}^{x}(|t|-t^2)e^{-t^2} dt$ for $x>0$,and $g(x)=\int_0^{x^2} \sqrt{t} e^{-t} dt$ for $x>0$.
$(1)$ 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 $\psi_2(x)=2x(\psi_1(\beta)-1)$
$(D)$ $f$ is an increasing function on the interval $[0, \frac{3}{2}]$
$(2)$ 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 $x \in(0, \frac{1}{2})$
$(D)$ $g(x) \leq \frac{2}{3}x^3-\frac{2}{5}x^5+\frac{1}{7}x^7$,for all $x \in(0, \frac{1}{2})$