Question

Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R.

Answer 1

$$\begin{gathered} f\left(x\right)=b\left(x+\cos x\right)+4 \\ f\text{'}\left(x\right)=b\left(1-\sin x\right) \\ \text{Given:}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) is decreasing on}\mathit{\text{R.}} \\ \Rightarrow f\text{'}\left(x\right)<0 \\ \Rightarrow b\left(1-\sin x\right)<0...\left(1\right) \\ \text{We know,} \\ \sin x\le 1 \\ \Rightarrow 1-\sin x\ge 0 \\ \Rightarrow b<0\left[\mathrm{Since}\left(1-\sin x\right)\ge 0,b\left(1-\sin x\right)<0\Rightarrow b<0\right] \\ \Rightarrow b\in \left(-\infty ,0\right) \end{gathered}$$