For the function\(f(x)=(\cos x)-x+1,x\in \mathrm{ℝ}\), between the following two statements \((S_1)\): \(f(x)=0\) for on…

For the function\(f(x)=(\cos x)-x+1,x\in \mathrm{ℝ}\), between the following two statements
\((S_1)\): \(f(x)=0\) for only one value of \(x\) in \([0, \pi]\).
\((S_2)\): \(f(x)\) is decreasing in \(\left[0,\frac{\pi }{2}\right]\) and increasing in \(\left[\frac{\pi }{2},\pi \right]\).

[JEE Main 2024, 8 Apr (Shift 1)]