Integral Calculus: JEE Advanced 2021 Mathematics 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 \[g(x)=\int_0^{x²} \sqrt{t} e^{-t} d t, x>0\]. Which of the following statements is TRUE?