Mark the correct alternative in the following:

Formula:- (i) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

Given:-

$f(x)=\cos |x|-2 a x+b$

$\frac{d(f(x))}{d x}=-\sin x-2 a=f^{\prime}(x)$

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

$\Rightarrow-\sin x-2 a>0$

$\Rightarrow 2 a<-\sin x$

$\Rightarrow 2 a \leq-1$

$\Rightarrow a \leq-\frac{1}{2}$