Let \( \psi_1 : [0,\infty) \to \mathbb{R},\; \psi_2 : [0,\infty) \to \mathbb{R},\; f : [0,\infty) \to \mathbb{R} \) and \( g : [0,\infty) \to \mathbb{R} \) be functions such that \( f(0)=g(0)=0 \),
\( \psi_1(x)=e^{-x}+x,\; \quad x\ge 0, \)
\( \psi_2(x)=x^2-2x-2e^{-x}+2,\; \quad x\ge 0, \)
\( f(x)=\displaystyle \int_{-x}^{x} (|t|-t^2)e^{-t^2}\,dt,\;\quad x>0 \)
and
\( g(x)=\displaystyle \int_{0}^{x^2} \sqrt{t}\,e^{-t}\,dt,\; \quad x>0 \)
Which of the following statements is TRUE?
