$2{x^3} - 6x + 5$ is an increasing function if

If $R$ is the least value of $a$ such that the function $f(x) = x^{2} + ax + 1$ is increasing on $[1, 2]$ and $S$ is the greatest value of $a$ such that the function $f(x) = x^{2} + ax + 1$ is decreasing on $[1, 2]$,then the value of $|R - S|$ is ..... .

If $y = 8x^3 - 60x^2 + 144x + 27$ is a decreasing function on the interval $(a, b)$,then $(a, b) = $

If $f(x) = xe^{x(1-x)}$,then $f(x)$ is...

Which statement among the following is true?
$(i)$ The function $f(x) = x|x|$ is strictly increasing on $R - \{0\}$.
$(ii)$ The function $f(x) = \log_{(1/4)} x$ is strictly increasing on $(0, \infty)$.
$(iii)$ $A$ one-one function is always an increasing function.
$(iv)$ $f(x) = x^{1/3}$ is strictly decreasing on $R$.

Which of the following is true about $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$?