Question

Write the set of values of k for which f(x) = kx − sin x is increasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=kx-\sin x \\ f\text{'}\left(x\right)=k-\cos x \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow k-\cos x>0 \\ \Rightarrow \cos x<k \\ \text{We know that the maximum value of cos}\mathit{\text{x}}\text{is 1.} \\ \text{Since cos}\mathit{\text{x}}\text{<}\mathit{\text{k,}}\text{the minimum value of}\mathit{\text{k}}\text{is 1.} \\ \mathit{\text{⇒}}k\in \left(1,\infty \right) \end{gathered}$$