Question

The function $f\left(x\right)=\cos x-2\lambda x$ is monotonically decreasing when

• A. $\lambda >\frac{1}{2}$
• B. $\lambda <\frac{1}{2}$
• C. $\lambda <2$
• D. $\lambda >2$

Answer 1

The correct option is B $\lambda <\frac{1}{2}$

$f\left(x\right)=\cos x-2\lambda x$
$f^{\prime }\left(x\right)=-\sin x-2\lambda =-(\sin x+2\lambda )$
Since, $-1\le \sin x\le 1$
$2\lambda -1\le f^{\prime }\left(x\right)\le 2\lambda +1$
For f(x) to be monotonically decreasing, f'(x)<0
$\Rightarrow 2\lambda -1<0$
$\Rightarrow \lambda <\frac{1}{2}$