Find the values of ' a ' for which $\displaystyle \mathrm{f}(x)=\sqrt{3} \sin x-\cos x-2 \mathrm{a} x+\mathrm{b}$ is decreasing on $\displaystyle \mathbb{R}$.

' $\displaystyle \mathrm{f}$ ' is a decreasing function iff $\displaystyle \mathrm{f}^{\prime}(x) \leq 0 \Rightarrow \sqrt{3} \cos x+\sin x-2 \mathrm{a} \leq 0$ \[\Rightarrow \sin \left(x+\frac{\pi}{3}\right) \leq \mathrm{a} \] We know that for all $\displaystyle x \in R,-1 \leq \sin \left(x+\frac{\pi}{3}\right) \leq 1$ \[\Rightarrow \mathrm{a} \in[1, \infty) \text { or }(1, \infty) \]