Consider the function $f\left(x\right)=x \cos x–\sin x$. Then identify the statement which is correct.

$f^{\mathrm{\prime }}\left(x\right)=-x\sin x=0$ when  $x=0 \text{or }\pi$

$\begin{array}{l}{f}^{\mathrm{\prime }}\left(0^{-}\right)=(-)(-)(-)<0\\ f^{\mathrm{\prime }}\left(0^{+}\right)=(-)(+)(+)<0\end{array}\}$  no sign change

This also implies that f  is decreasing at $x=0$ ; Thus, (b) is correct. $f"\left(x\right)=–(x\cos x+\sin x)$

$f^{\prime \prime }\left(\pi \right)=-(-\pi )>0$ , i.e.,minima at   $x=\pi$

$\Rightarrow f^{\prime \prime }(-\pi )=-(-\pi )<0$,  i.e.,maxima at   $x=-\pi$