For what interval is the function $f(x) = \sin x - \cos x$ strictly increasing?

The function $f(x) = \tan x - 4x$ is strictly decreasing on $\rule{1cm}{0.15mm}$

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?

For what values of $x$ is the function $f(x) = x^3 - 3x^2 - 9x + 22$ strictly decreasing?

The function $f(x) = \sin^4 x + \cos^4 x$ is increasing in

Let $f(x) = \int\limits_1^x {\left( {t\ln(t) - \frac{{\ln(t)}}{t}} \right)dt}$ for $x > 1$. Then: