Question

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

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

Answer 1

The correct option is B

$p>\frac{1}{2}$

Explanation for correct option

$f\left(x\right)$ is monotonically decreasing for $f\text{'}\left(x\right)<0$

Given $f\left(x\right)=\cos \left(x\right)-2px$

$\begin{array}{rcl}\Rightarrow f\text{'}\left(x\right)& =& -\sin \left(x\right)-2p\end{array}$

$f\text{'}\left(x\right)<0$ for monotonically decreasing

$$\begin{gathered} \Rightarrow -\sin \left(x\right)-2p<0 \\ \Rightarrow \frac{1}{2}(\sin \left(x\right)+p)>0 \\ \Rightarrow p>-\frac{1}{2}(\sin x\in [-11]) \end{gathered}$$

Hence, the correct option is $\left(\mathrm{B}\right)$