If $f(x) = \sin x - \cos x$,$0 \leq x \leq 2\pi$,then $f(x)$ is strictly decreasing in the interval:

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

For what value of $a$ is the function $f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1$ a decreasing function for all $x \in R$?

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?

Prove that the function $f$ given by $f(x) = \log |\cos x|$ is decreasing on $\left(0, \frac{\pi}{2}\right)$ and increasing on $\left(\frac{3\pi}{2}, 2\pi\right)$.

Let $f(x) = \int e^x (x - 1)(x - 2) dx$. In which interval is $f$ a decreasing function?