For what values of $x$ is the function $f(x) = \tan^{-1}(\sin x + \cos x)$ strictly increasing?

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

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,

If a number is drawn at random from the set {$1, 3, 5, 7, \dots, 59$},then the probability that it lies in the interval in which the function $f(x) = x^3 - 16x^2 + 20x - 5$ is strictly decreasing,is

The function $f(x) = x^4 - 4x$ is decreasing in the interval

In which interval does the function $f(x) = 2x^2 - \log |x|$ $(x \neq 0)$ monotonically increase?