Observe the following statements:
$A: f(x) = 2x^3 - 9x^2 + 12x - 3$ is increasing outside the interval $(1, 2)$.
$R: f^{\prime}(x) < 0$ for $x \in (1, 2)$.
Then,which of the following is true?

If $f(x)=x^3+b x^2+c x+d$ and $0 < b^2 < c$,then in $(-\infty, \infty)$

Let $g(x) = 2f(x/2) + f(1 - x)$ and $f''(x) < 0$ for $0 \le x \le 1$. Then $g(x)$:

If $f(x) = x^3 - x^2 + 100x + 1001$,then:

The function $f(x)=2 x^3-9 x^2+12 x+29$ is monotonically increasing 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,