If $f(x)=\sqrt{3} \sin x-\cos x-2 a x+b$ decreases for all values of $x$, then

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Aicharya · Accepted Answer

$f(x)=\sqrt{3} \sin x-\cos x-2 a x+b$ $\Rightarrow f^{\prime}(x)=\sqrt{3} \cos x+\sin x-2 a$ $\begin{aligned} & =2\left(\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin xight)-2 a \ & =2 \sin \left(x+\frac{\pi}{3}ight)-2 a\end{aligned}$ When, $f(x)$ decreases. Then, $f^{\prime}(x) \leq 0$ $\begin{aligned} & \Rightarrow \quad 2 \sin \left(x+\frac{\pi}{3}ight)-2 a \leq 0 \ & \Rightarrow \quad 2 a \geq 2 \sin \left(x+\frac{\pi}{3}ight) \Rightarrow a \geq \sin \left(x+\frac{\pi}{3}ight)\end{aligned}$ $\begin{aligned} & \because \quad \sin \left(x+\frac{\pi}{3}ight) \leq 1 \ & 	herefore \quad a \geq 1\end{aligned}$

If $f(x)=\sqrt{3} \sin x-\cos x-2 a x+b$ decreases for all values of $x$, then

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