Question

Show that f(x) = x − sin x is increasing for all x ∈ R.

Answer 1

$$\begin{gathered} f\left(x\right)=x-\sin x \\ f\text{'}\left(x\right)=1-\cos x \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow 1-\cos x>0 \\ \Rightarrow f\text{'}\left(x\right)⩾0\mathrm{for}\mathrm{all}x\in R\left[\because \mathrm{Cos}x⩽1\right] \\ \mathrm{So},f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{for}\mathrm{all}x\in R. \end{gathered}$$