Question

Find 'a' for which f(x) = a (x + sin x) + a is increasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=a\left(x+\sin x\right)+a \\ f\text{'}\left(x\right)=a\left(1+\cos x\right) \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow a\left(1+\cos x\right)>0...\left(1\right) \\ \text{We know,} \\ -1\le \cos x\le 1,\forall x\in R \\ \Rightarrow 0\le \left(1+\cos x\right)\le 2,\forall x\in R \end{gathered}$$


$$\begin{gathered} \therefore a>0\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ \Rightarrow a\in \left(0,\infty \right) \end{gathered}$$