If $f(x) = x$,$-1 \le x \le 1$,then the function $f(x)$ is

The function $f(x) = \frac{\lambda \sin x + 6 \cos x}{2 \sin x + 3 \cos x}$ is monotonically increasing,if:

The function $f(x) = \cot^{-1} x + x$ is increasing in the interval.

For what value of $x$ is the function $f(x) = e^{ax} + e^{-ax}$,where $a > 0$,an increasing function?

For what values of $x$ is the function $f(x) = x - \log x$ strictly decreasing?

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,