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