The function $f(x) = x^3 - 3x^2 - 24x + 5$ is an increasing function in the interval given below:

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

On which of the following intervals is the function $f$ given by $f(x)=x^{100}+\sin x-1$ decreasing?

The function $f(x) = x + \cos x$ is

Function $f(x) = |\sin x|$,$x \in \left(-\frac{\pi}{2}, 0\right)$ is . . . . . . .

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