Question:
Mark the correct alternative in the following:
Function $f(x)=\cos x-2 \lambda x$ is monotonic decreasing when
A. $\lambda>\frac{1}{2}$
B. $\lambda<\frac{1}{2}$
C. $\lambda<2$
D. $\lambda>2$
Solution:
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 $f^{\prime}(x)<0$
$-\sin x-2 \lambda<0$
$\Rightarrow \operatorname{Sin} x+2 \lambda>0$
$\Rightarrow 2 \lambda>-\sin x$
$\Rightarrow 2 \lambda>1$
$\Rightarrow \lambda>\frac{1}{2}$