Question

Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R.

Answer 1

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