If $f(x) = \frac{x}{\sin x}$ and $g(x) = \frac{x}{\tan x}$,where $0 < x \le 1$,then in this interval:

In which of the following intervals does $f(x) = \sin x$ increase less rapidly than $g(x) = \cos x$?

Let $g(x) = 2f(\frac{x}{2}) + f(2 - x)$ and $f''(x) < 0$ for all $x \in (0, 2)$. Then $g(x)$ increases in

For what value of $x$ is the function $f(x) = e^{ax} + e^{-ax}$,where $a > 0$,an increasing function?

Let $f : R \rightarrow R$ be a twice differentiable function such that $f^{\prime\prime}(x) > 0$ for all $x \in R$ and $f^{\prime}(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^{2}x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$. Consider the following two statements:
$(I)$ $g$ is increasing in $(0, \frac{\pi}{4})$
$(II)$ $g$ is decreasing in $(\frac{\pi}{4}, \frac{\pi}{2})$
Then,

The interval of the decreasing function $f(x) = x^3 - x^2 - x - 4$ is