Question

Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R.

Answer 1

$$\begin{gathered} f\left(x\right)=\sin x-bx+c \\ f\text{'}\left(x\right)=\cos x-b \\ \text{Given:}f\left(x\right)\text{is decreasing on}\mathit{\text{R.}} \\ f\text{'}\left(x\right)<0,\forall x\in R \\ \Rightarrow \cos x-b<0,\forall x\in R \\ \text{⇒}\cos x-b<0,\forall x\in R \\ \Rightarrow \cos x<b,\forall x\in R \\ \Rightarrow b⩾1\left[\because -1⩽\cos x⩽1\right] \end{gathered}$$