Practicing Success
If the function $f(x)=3 \cos |x|-6 a x+b$ increases for all $x \in R$, then the range of values of a is given by |
$(-1 / 2, \infty)$ $(-\infty,-1 / 2)$ $(-\infty,-2)$ $(-2, \infty)$ |
$(-\infty,-1 / 2)$ |
We have, $f(x)=3 \cos |x|-6 a x+b$ $\Rightarrow f(x)=3 \cos x-6 a x+b$ [∵ cos |x| = cos x for all x] $\Rightarrow f'(x) =-3 \sin x-6 a$ For f(x) to be increasing on R, we must have $f'(x)>0$ for all $x \in R$ $\Rightarrow -3 \sin x-6 a>0$ for all $x \in R$ $\Rightarrow \sin x+2 a<0$ for all $x \in R$ $\Rightarrow \sin x<-2 a$ for all $x \in R$ $\Rightarrow 1>-2 a$ [∵ Max. value of sin x is 1] $\Rightarrow a<-\frac{1}{2}$ $\Rightarrow a \in(-\infty,-1 / 2)$ |