The function $f(x) = x^4 - 4x$ is decreasing in the interval

The function $f(x) = 2\ln|x| - x|x|$ is increasing on the interval

If $x \in (0, 1)$,then what type of function is $f(x) = x^{100} + \sin x - 1$?

Consider the following statements $S$ and $R$:
$S$: Both $\sin x$ and $\cos x$ are decreasing functions in $\left( \frac{\pi}{2}, \pi \right)$.
$R$: If a differentiable function decreases in $(a, b)$,then its derivative also decreases in $(a, b)$.
Which of the following is true?

If $f(x) = \int_{x^2}^{x^2 + 1} e^{-t^2} dt$,then $f(x)$ increases in

If $f(x) = \frac{\log x}{x}$ $(x > 0)$,then it is increasing in