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Question:

Show that for ³ 1, (x) = √3 sin x – cos x – 2ax + b is decreasing in R.

Solution:

Given,

(x) = √3 sin x – cos x – 2ax + b, a ³ 1

On differentiating both sides w.r.t. x, we get

f’ (x) = √3 cos x + sin x – 2a

For increasing function, f’ (x) < 0

So,    $\sqrt{3} \cos x+\sin x-2 a<0$

$2\left(\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x\right)-2 a<0$

$\frac{\sqrt{3}}{2} \cos x+\frac{1}{2} \sin x-a<0$

$\left(\cos \frac{\pi}{6} \cos x+\sin \frac{\pi}{6} \sin x\right)-a<0$

$\cos \left(x-\frac{\pi}{6}\right)-a<0$

As $\cos x \in[-1,1]$ and $a \geq 1$

Hence, $f^{\prime}(x)<0$

Therefore, the given function is decreasing in R.

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