Question:

Which must be true for: I) For $x\in\mathbb{R}$, let $\tan^{-1}x\in(-\pi/2,\pi/2)$. Then minimum value of $f(x)=\int_0^{x\tan^{-1}x}\frac{e^{t-\cos t}}{1+t^{2023}}dt$ is zero. II) If $f(x)$ is periodic with period $T$, then $\int_{a+nT}^{b+nT}f(x)dx=\int_a^b f(x)dx$ for $n\in\mathbb{Z}$. III) If $\psi(x)\le\phi(x)$ for $a\le x\le b$, then $\int_a^b|\psi(x)|dx\le\int_a^b|\phi(x)|dx$.