Let $y = f(x)$ and $y = g(x)$ be two differentiable functions in $[0, 2]$ such that $f(0) = 3$,$f(2) = 5$,$g(0) = 1$,and $g(2) = 2$. If there exists at least one $c \in (0, 2)$ such that $f'(c) = k g'(c)$,then $k$ must be:

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

The value of $\left[ \frac{\log (x/e)}{x - e} \right]$ for all $x > e$ is equal to (where $[.]$ denotes the greatest integer function).

Let $f:[a, b] \rightarrow R$ be such that $f$ is differentiable in $(a, b)$, continuous at $x=a$ and $x=b$, and $f(a)=0=f(b)$. Then:

Let $f(x) = \begin{cases} x^\alpha \ln x, & x > 0 \\ 0, & x = 0 \end{cases}$. Rolle's theorem is applicable to $f$ for $x \in [0, 1]$ if $\alpha = $

If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is: