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Question Number 114765 by dw last updated on 21/Sep/20

$$Findthe\mathit{m}\mathit{a}\mathit{x}\mathit{i}\mathit{m}\mathit{u}\mathit{m}.and\mathit{m}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{u}\mathit{m}valueof\lfloor 1+sinx\rfloor +\lfloor 1+sin3x\rfloor +\lfloor 1+sin2x\rfloor$$$$$$$$$$

Answered by PRITHWISH SEN 2 last updated on 21/Sep/20

$$\mathrm{let}$$$$\mathrm{f}\left(\mathrm{x}\right)=\lfloor 1+\sin \mathrm{x}\rfloor +\lfloor 1+\sin 2\mathrm{x}\rfloor +\lfloor 1+\sin 3\mathrm{x}\rfloor$$$$\mathrm{Now},$$$$-1⩽\sin \mathrm{nx}⩽1$$$$0⩽\lfloor 1+\sin \mathrm{nx}\rfloor ⩽2$$$$\therefore \mathbf{the}\mathbf{min}.\mathbf{value}\mathbf{of}\mathbf{f}\left(\mathbf{x}\right)=0$$$$\mathbf{now}\mathbf{for}\mathbf{the}\mathbf{max}.\mathbf{value}$$$$\mathrm{the}\mathrm{period}\mathrm{of}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{L}.\mathrm{C}.\mathrm{M}(2\pi ,\frac{2\pi }{2},\frac{2\pi }{3})=2\pi$$$$\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{fundamental}\mathrm{period}\mathrm{of}$$$$\sin \mathrm{x}\in [-\frac{\pi }{2},\frac{\pi }{2}]$$$$\therefore \mathbf{the}\mathbf{max}.\mathbf{value}\mathbf{of}$$$$\mathbf{f}\left(\mathbf{x}\right)=\mathbf{max}.\{\mathbf{f}\left(\frac{\mathit{\pi }}{2}\right),\mathbf{f}\left(\frac{\mathit{\pi }}{4}\right),\mathbf{f}\left(\frac{\mathit{\pi }}{6}\right)\}$$$$\{\because \mathbf{sinx}\mathbf{is}\mathbf{an}\mathbf{increasing}\mathbf{function}\mathbf{in}[-\frac{\mathit{\pi }}{2},\frac{\mathit{\pi }}{2}]\}$$$$\mathbf{f}\left(\frac{\mathit{\pi }}{2}\right)=\lfloor 1+\sin \frac{\pi }{2}\rfloor +\lfloor 1+\sin \frac{2\pi }{2}\rfloor +\lfloor 1+\sin \frac{3\pi }{2}\rfloor$$$$=2+1+0=3$$$$\mathrm{f}\left(\frac{\pi }{4}\right)=\lfloor 1+\sin \frac{\pi }{4}\rfloor +\lfloor 1+\sin \frac{2\pi }{4}\rfloor +\lfloor 1+\sin \frac{3\pi }{4}\rfloor$$$$=1+2+1=4$$$$\mathrm{f}\left(\frac{\pi }{6}\right)=\lfloor 1+\sin \frac{\pi }{6}\rfloor +\lfloor 1+\sin \frac{2\pi }{6}\rfloor +\lfloor 1+\sin \frac{3\pi }{6}\rfloor$$$$=1+1+2=4$$$$\therefore \mathbf{the}\mathbf{max}.\mathbf{value}\mathbf{of}\mathbf{f}\left(\mathbf{x}\right)=4$$

Commented by dw last updated on 21/Sep/20

$$ThankyouSir$$

Commented by PRITHWISH SEN 2 last updated on 21/Sep/20

$$\mathrm{welcome}$$

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