Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 33330 by artibunja last updated on 14/Apr/18

Answered by MJS last updated on 15/Apr/18

$$\mid 3^{\tan \pi x}-\frac{3}{3^{\tan \pi x}}\mid ⩾2$$$$\mid \frac{3^{2\tan \pi x}-3}{3^{\tan \pi x}}\mid ⩾2$$$$\mid 3^{2\tan \pi x}-3\mid ⩾2\times 3^{\tan \pi x}$$$$3^{\tan \pi x}=t$$$$$$$$1.(t^2-3)⩾0$$$$t^2-3⩾2t$$$$t^2-2t-3⩾0$$$$t⩽-1\vee t⩾3$$$$3^{\tan \pi x}⩾0\mathrm{for}x\in \mathbb{R}\Rightarrow 3^{\tan \pi x}⩾3\Rightarrow$$$$\Rightarrow \tan \pi x⩾1$$$$$$$$2.(t^2-3)<0$$$$t^2-3<2t$$$$t^2-2t-3<0$$$$-1<t<3$$$$3^{\tan \pi x}⩾0\mathrm{for}x\in \mathbb{R}\Rightarrow 0⩽3^{\tan \pi x}<3\Rightarrow$$$$\Rightarrow -\mathrm{\infty }⩽\tan \pi x<1\Rightarrow \tan \pi x<1$$$$$$$$\mid 3^{\tan \pi x}-3^{1-\tan \pi x}\mid ⩾2\mathrm{is}\mathrm{true}\mathrm{for}$$$$(\tan \pi x⩾1)\vee (\tan \pi x<1)\Rightarrow$$$$\Rightarrow \mathrm{true}\mathrm{for}\mathrm{all}x\in \mathbb{R}$$

Answered by artibunja last updated on 17/Apr/18

Terms of Service

Privacy Policy

Contact: info@tinkutara.com