In which interval is the function $f(x) = x^2 - x + 1$ not monotonic?

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?

If $f(x) = x e^{x(1-x)}, x \in R$,then $f(x)$ is

The function $f(x) = \frac{\lambda \sin x + 6 \cos x}{2 \sin x + 3 \cos x}$ is monotonically increasing,if:

The function $\sin x - bx + c$ will be increasing in the interval $(-\infty, \infty)$,if

Prove that the function $f$ given by $f(x)=x^{2}-x+1$ is neither strictly increasing nor decreasing on $(-1,1).$