For all twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R},$ with $f(0)=f(1)=f^{\prime}(0)=0,$ which of the following is true?

If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

The value of $c$ for which the Mean Value Theorem holds for the function $f(x) = \log_{e} x$ on the interval $[1, 3]$ is:

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

$A$ value of $c$ for which the conclusion of the Mean Value Theorem holds for the function $f(x) = \log_e x$ on the interval $[1, 3]$ is

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