The function $f(x) = x + \frac{1}{x}, (x \neq 0)$ is a non-increasing function in the interval

If the maximum value of $a$,for which the function $f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$,is $\bar{a}$,then $f_{\bar{a}}\left(\frac{\pi}{8}\right)$ is equal to

Prove that the logarithmic function $f(x) = \log x$ is strictly increasing on $(0, \infty).$

For what values of $x$ is the function $f(x) = x^9 + 3x^7 + 64$ strictly increasing?

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 $f(x)=x^3+a x^2+b x+5 \sin ^2 x$ is an increasing function on $R$,then