Question:
Mark the correct alternative in the following:
If the function $f(x)=\cos |x|-2 a x+b$ increases along the entire number scale, then
A. $a=b$
B. $a=\frac{1}{2} b$
C. $a \leq-\frac{1}{2}$
D. $a>-\frac{3}{2}$
Solution:
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 $f^{\prime}(x)>0$
$\Rightarrow-\sin x-2 a>0$
$\Rightarrow 2 a<-\sin x$
$\Rightarrow 2 a \leq-1$
$\Rightarrow a \leq-\frac{1}{2}$