If $f(x) = x^3 + bx^2 + cx + d$ and $0 < b^2 < c$,then on $R$,$f(x)$ is:

The function $f(x)=3x^{4}+16x^{3}-30x^{2}+10$ is increasing for

For the function $f(x) = \cos x - x + 1, x \in R$,consider the following two statements:
$(S1)$ $f(x) = 0$ for only one value of $x$ in $[0, \pi]$.
$(S2)$ $f(x)$ is decreasing in $[0, \frac{\pi}{2}]$ and increasing in $[\frac{\pi}{2}, \pi]$.

Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is
$(a)$ increasing
$(b)$ decreasing

What type of function is $f(x) = \frac{x - 2}{x + 1}$,where $x \neq -1$?

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?