Question

Show that f(x) = x + cos x − a is an increasing function on R for all values of a.

Answer 1

$$\begin{gathered} f\left(x\right)=x+\cos x-a \\ f\text{'}\left(x\right)=1-\sin x \\ \text{We know,} \\ \sin x\le 1,\forall x\in R \\ \Rightarrow -\sin x\ge -1,\forall x\in R \\ \Rightarrow 1-\sin x\ge 0,\forall x\in R \\ \Rightarrow f\text{'}\left(x\right)\ge 0,\forall x\in R \\ \mathrm{Hence}\text{,}f\left(x\right)\text{is increasing on}\mathit{\text{R}}\text{for all values of}a. \end{gathered}$$