$f(x) = \tan^{-1} x - x$ is . . . . . . ,$x \in R$.

In which interval is the function $f(x) = 2x^2 - \ln |x|$ $(x \ne 0)$ monotonically decreasing?

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 $[\frac{\pi}{2}, \pi]$.

The range of values of $x$ for which $f(x)=x^3+6x^2-36x+7$ is increasing 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 $R$.

For what values of $K$ is the function $f(x) = Kx^3 + 9x^2 + 9x + 3$ an increasing function on $R$?