If a function $f(x)$ is increasing in an interval $x \in [a, b]$,then which of the following will always be correct?

If $f(x) = x^3 + bx^2 + cx + d$ and $0 < b^2 < c$,then on $R$,$f(x)$ is:

If $f(x) = \frac{x}{\sin x}$ and $g(x) = \frac{x}{\tan x}$,where $0 < x \le 1$,then in this 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,

Function $f(x) = \log_{10} \cos x$ is . . . . . . function in the interval $\left(0, \frac{\pi}{2}\right)$.

The function $f(x) = 2x^3 - 3x^2 + 90x + 174$ is increasing in the interval