Question

The function $f\left(x\right)=\cos |x|-2ax+b$ increases along the entire number scale. The range of ${}^{\prime }{a}^{\prime }$ is given by-

• A. $a=b$
• B. $a=\frac{b}{2}$
• C. $a\le -\frac{1}{2}$
• D. $a\le -\frac{3}{2}$

Answer 1

The correct option is C $a\le -\frac{1}{2}$

Now
$\cos \left(|x|\right)=\cos \left(x\right)$
for all real $x$.
And
$f^{\prime }\left(x\right)\ge 0$ for $y$ to be an increasing function for all real $x.$
Hence
$\sin x-2a\ge 0$
or
$a\le \frac{\sin x}{2}$
Now
$\sin xϵ[-1,1]$
Hence
$a\le \frac{1}{2}$ and $a\le \frac{-1}{2}$
Hence from the above two relations, we get
$a\le \frac{-1}{2}$