JEE Math Practice Question

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 \[\begin{aligned} & f(0)=\mathrm{g}(0)=0, \\ & \psi_1(\mathrm{x})=\mathrm{e}^{-\mathrm{x}}+\mathrm{x}, \quad \mathrm{x} \geq 0, \\ & \psi_2(\mathrm{x})=\mathrm{x}^2-2 \mathrm{x}-2 \mathrm{e}^{-\mathrm{x}}+2, \mathrm{x} \geq 0, \\ & f(\mathrm{x})=\int_{-\mathrm{x}}^{\mathrm{x}}\left(|\mathrm{t}|-\mathrm{t}^2\right) \mathrm{e}^{-\mathrm{t}^2} \mathrm{dt}, \mathrm{x}>0 \end{aligned}\] and \[g(x)=\int_0^{x^2} \sqrt{t} e^{-t} d t, x>0\]. Which of the following statements is TRUE?

1. A.$f(\sqrt{\ln 3})+g(\sqrt{\ln 3})=\frac{1}{3}$
2. B.For every $x>1$, there exists an $\alpha \in(1, x)$ such that $\psi_{1}(x)=1+\alpha x$
3. C.For every $x>0$, there exists a $\beta \in(0, x)$ such that $\psi_{2}(x)=2 x\left(\psi_{1}(\beta)-1\right)$
4. D.$f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$