The function $f(x) = e^{-1/x}$ is strictly increasing for all $x$ where

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?

If $f(x)=x^x$,then the interval in which $f(x)$ decreases is

In which interval is the function $f(x) = \tan^{-1}(\sin x + \cos x)$ an increasing function?

If $f(x) = x^2 + kx + 1$ is a strictly increasing function on the interval $[1, 2]$,then what is the minimum value of $k$?

Function $f(x) = \frac{\lambda \sin x + 3 \cos x}{2 \sin x + 6 \cos x}$ is monotonic increasing,then :