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Question Number 34029 by rahul 19 last updated on 29/Apr/18

$$\mathit{N}umberofintegralvaluesofxfor$$ $$which$$ $$\frac{(\frac{\pi }{2^{{\tan }^{-1}x}}-4)(x-4)(x-10)}{x!-(x-1)!}<0$$

Commented byrahul 19 last updated on 29/Apr/18

$$x!=1\times 2\times 3.........\times (x-1)\times x.$$

Answered by MJS last updated on 29/Apr/18

$$f\left(x\right)=\frac{(\frac{\pi }{2^{{\tan }^{-1}x}}-4)(x-4)(x-10)}{x!-(x-1)!}<0$$ $$$$ $$x!-(x-1)!>0\mathrm{\forall }x\in \mathbb{N}$$ $$\Rightarrow (\frac{\pi }{2^{{\tan }^{-1}x}}-4)(x-4)(x-10)<0$$ $$$$ $$\frac{\pi }{2^{{\tan }^{-1}x}}-4=\frac{\pi -4\times 2^{{\tan }^{-1}x}}{2^{{\tan }^{-1}x}}$$ $${\tan }^{-1}x>0\Rightarrow 2^{{\tan }^{-1}x}>1\Rightarrow \frac{\pi -4\times 2^{{\tan }^{-1}x}}{2^{{\tan }^{-1}x}}<0\mathrm{\forall }x\in \mathbb{N}$$ $$\Rightarrow (x-4)(x-10)>0$$ $$$$ $$x-4<0\wedge x-10<0\Rightarrow x<4$$ $$x-4>0\wedge x-10>0\Rightarrow x>10$$ $$$$ $$f\left(x\right)⩾0\Rightarrow 4⩽x⩽10$$ $$f\left(x\right)\mathrm{is}\mathrm{not}\mathrm{defined}\mathrm{for}x=0\wedge x=1$$ $$f\left(x\right)<0\Rightarrow x\in \{2;3\}\cup \{x\in \mathbb{N}\mid x>10\}$$

Commented byrahul 19 last updated on 29/Apr/18

$$Youare\mathrm{getting}\mathrm{this}\mathrm{answer}\mathrm{after}$$ $$\mathrm{assuming}{\tan }^{-1}x>0,right?$$ $$whatif{it}^{\prime }s<0?$$

Commented byMJS last updated on 29/Apr/18

$${\tan }^{-1}x⩾0\mathrm{for}\mathrm{all}x⩾0$$ $$\mathrm{and}x!\mathrm{is}\mathrm{not}\mathrm{defined}\mathrm{for}x<0$$

Commented byrahul 19 last updated on 29/Apr/18

$$Ohh,yes!$$ $$\mathcal{T}hankyousir.$$

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