Let f:[0,2] rightarrow R be the function defi | vedclass

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Question

Let $\pi x-\frac{\pi}{4}=	heta \in\left[\frac{-\pi}{4}, \frac{7 \pi}{4}ight]$

$	ext { So, } \quad\left(3-\sin \left(\frac{\pi}{2}+2 	hetaigh

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Let $\pi x-\frac{\pi}{4}=	heta \in\left[\frac{-\pi}{4}, \frac{7 \pi}{4}ight]$

$	ext { So, } \quad\left(3-\sin \left(\frac{\pi}{2}+2 	hetaight)ight) \sin 	heta \geq \sin (\pi+3 	heta)$

$\Rightarrow \quad(3-\cos 2 	heta) \sin 	heta \geq-\sin 3 	heta$

$\Rightarrow \quad \sin 	heta\left[3-4 \sin ^2 	heta+3-\cos 2 	hetaight] \geq 0$

$\Rightarrow \quad \sin 	heta(6-2(1-\cos 2 	heta)-\cos 2 	heta) \geq 0$

$\Rightarrow \quad \sin 	heta(4+\cos 2 	heta) \geq 0$

$\Rightarrow \quad \sin 	heta \geq 0$

$\Rightarrow \quad 	heta \in[0, \pi] \Rightarrow 0 \leq \pi x-\frac{\pi}{4} \leq \pi$

$\Rightarrow \quad x \in\left[\frac{1}{4}, \frac{5}{4}ight]$

$\Rightarrow \quad \beta-\alpha=1$

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