Find the values of' a 'for which $\displaystyle \mathrm{f}(\mathrm{x})=\sin \mathrm{x}-\mathrm{a} \mathrm{x}+\mathrm{b}$ is increasing on R.

\[\mathrm{f}^{\prime}(\mathrm{x})=\cos \mathrm{x}-\mathrm{a} \] For $\displaystyle \mathrm{f}(\mathrm{x})$ to be increasing, $\displaystyle \mathrm{f}^{\prime}(\mathrm{x}) \geq 0$ i.e., $\displaystyle \cos \mathrm{x} \geq \mathrm{a}$ Since, $\displaystyle -1 \leq \cos \mathrm{x} \leq 1$ \[\Rightarrow \mathrm{a} \leq-1 \] Hence, $\displaystyle \mathrm{a} \in(-\infty,-1]$. (Also, accept $\displaystyle \mathrm{a} \in(-\infty,-1)$ )