Question

The function $f\left(x\right)=cosx-2px$ is monotonically decreasing for

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

Answer 1

Answer: (B)

Given $f\left(x\right)=cosx-2px$ .
Differentiating $f\left(x\right)$ with respect to $x$ , we get
$f^{{}^{\prime }}\left(x\right)=-sinx-2p$
For $f\left(x\right)$ to be monotonically decreasing, $f^{{}^{\prime }}\left(x\right)<0$ .
$\Rightarrow -sinx-2p<0$
$\Rightarrow p>-\frac{1}{2}sinx..........\left(i\right)$
Since, $-1\le sinx\le 1$
$\Rightarrow -\frac{1}{2}\le -\frac{1}{2}sinx\le \frac{1}{2}..........\left(ii\right)$
From equations $\left(i\right)\&\left(ii\right)$ , we get
$p>\frac{1}{2}$