A function $f$ is defined by $f\left(x\right)={\int }_0^{\pi }\mathrm{c}\mathrm{o}\mathrm{s}t \mathrm{c}\mathrm{o}\mathrm{s}\left(x-t\right)dt, 0\le x\le 2\mathrm{\pi }$, then which of the following hold(s) good

Minimum value of $f$ is $-\frac{\pi }{2}$

Maximum value of $f$ is $\pi$

There exists atleast one $c\in \left(\mathrm{0,} 2\pi \right)$ it $f^{\text{'}}\left(c\right)=0$

$f\left(x\right)$ is continuous but not differentiable in $\left(0, 2\mathrm{\pi }\right)$