The value of the Rolle's theorem for the function $f(x)=2 \sin x+\sin 2 x$ in the interval $[0, \pi]$ is

If the function $y=g(x)$ representing the slopes of the tangents drawn to the curve $y=3 x^4-5 x^3-12 x^2+18 x+3$ is strictly increasing, then the domain of $g(x)$ is

Consider the following functions

I. $f(x)= \begin{cases}\frac{1}{2}-x & , x<\frac{1}{2} \\ \left(\frac{1}{2}-x\right)^2 & , x \geq \frac{1}{2}\end{cases}$

II. $f(x)=|3 x-1|$

III. $f(x)=x|x|$

IV. $f(x)=|x|$

Then, on $[0,1]$ Lagrange's mean value theorem is applicable to the functions

$$ \int \frac{e^{\sin x}(\sin 2 x-8 \cos x)}{2(\sin x-3)^2} d x= $$