Question

Function f(x) = cos x − 2 λ x is monotonic decreasing when
(a) λ > 1/2
(b) λ < 1/2
(c) λ < 2
(d) λ > 2

Answer 1

(a) λ > 1/2

$$\begin{gathered} f\left(x\right)=\cos x-2\lambda x \\ f\text{'}\left(x\right)=-\sin x-2\lambda \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be decreasing, we must have} \\ f\text{'}\left(x\right)<0 \\ \Rightarrow -\sin x-2\lambda <0 \\ \Rightarrow s\mathrm{in}x+2\lambda >0 \\ \Rightarrow 2\lambda >-\sin x \\ \text{We know that the maximum value of -sin}\mathit{\text{x}}\text{is 1.} \\ \Rightarrow 2\mathrm{\lambda }>1 \\ \Rightarrow \mathrm{\lambda }>\frac{1}{2} \end{gathered}$$