If $f(x)=x^3+b x^2+c x+d$ and $0 < b^2 < c$,then in $(-\infty, \infty)$

The function $f(x) = x + \frac{1}{x}, (x \neq 0)$ is a non-increasing function in the interval

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right)$,where $x \in\left[0, \frac{\pi}{2}\right]$,consider the following two statements:
$(I)$ $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.
$(II)$ $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
Which of the following is correct?

The function $f(x) = 2x^3 - 6x + 5$ is an increasing function,if

The function $f(x)=4 \sin ^3 x-6 \sin ^2 x+12 \sin x+100$ is strictly

Consider two statements $S_1$ and $S_2$.
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?