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

Consider two statements $S_1$ and $S_2$.
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?

$F(x) = \log |\sin x|$,where $x \in (0, \pi)$,is strictly increasing on

Let $f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+19$. Then, $f(x)=0$ has

If $f(x) = \frac{a \sin x + b \cos x}{c \sin x + d \cos x}$ is decreasing for all $x$,then

In which interval is the function $f(x) = x^x$ a decreasing function?