Mark the correct alternative in the following:

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

Given:-

$f(x)=\cos x-2 \lambda x$

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

for decreasing function $f^{\prime}(x)<0$

$-\sin x-2 \lambda<0$

$\Rightarrow \operatorname{Sin} x+2 \lambda>0$

$\Rightarrow 2 \lambda>-\sin x$

$\Rightarrow 2 \lambda>1$

$\Rightarrow \lambda>\frac{1}{2}$