If $y = ax^3 + 3x^2 + (2a + 1)x + 1000$ is a strictly increasing function for all values of $x$,then:

If $f(x) = \frac{1}{x + 1} - \log(1 + x)$,where $x > 0$,then what type of function is $f$?

If $f(x)=\sqrt{3} \sin x-\cos x-2 a x+b$ decreases for all values of $x$,then

The function $f(x) = 1 - e^{-\frac{x^2}{2}}$ is .......

The function $y = 2x^3 - 9x^2 + 12x - 6$ is monotonically decreasing when

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 $R$.