$$\begin{aligned} & f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right] \\ & f^{\prime}(x)=\cos x+3-\frac{2}{\pi}(2 x+1) \\ & =\cos x-\frac{4 x}{\pi}-\frac{2}{\pi}+3 \\ & \text { as } x \in\left[0, \frac{\pi}{2}\right] \\ & \frac{4 x}{\pi} \in[0,2] \end{aligned}$$

$$\Rightarrow 3-\frac{2}{\pi}-\frac{4 x}{\pi}>0$$

and also $$\cos x>0$$ when $$x \in\left[0, \frac{\pi}{2}\right]$$

$$\Rightarrow f^{\prime}(x)>0$$

$$\Rightarrow f(x)$$ is increasing

Now, $$f^{\prime \prime}(x)=-\sin x-\frac{4}{\pi}<0 \forall x \in\left[0, \frac{\pi}{2}\right]$$

Hence, $$f^{\prime}(x)$$ is decreasing

$$\therefore \quad$$ Both statements (I) and (II) are true