The interval for which the given function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is decreasing,is

If $f(x) = 1 + x + \int_{1}^{x} (\ln^2 t + 2 \ln t) \, dt$,then $f(x)$ increases in

If $f(x) = \int_{x^2}^{x^2 + 1} e^{-t^2} dt$,then $f(x)$ increases in

The function $f(x)=\frac{x}{3}+\frac{3}{x}$ decreases in the interval

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right)$,where $x \in\left[0, \frac{\pi}{2}\right]$,consider the following two statements:
$(I)$ $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.
$(II)$ $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.
Which of the following is correct?

Find the intervals in which the function $f$ given by $f(x) = 4x^3 - 6x^2 - 72x + 30$ is
$(a)$ increasing
$(b)$ decreasing.