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Question Number 110779 by ZiYangLee last updated on 30/Aug/20

$$\mathrm{If}\:\mathrm{0}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}, \\$$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{2}}{\pi}{x}\leqslant\mathrm{sin}\:{x}\leqslant{x} \\$$$$\mathrm{without}\:\mathrm{graphical}\:\mathrm{method}. \\$$

Answered by Her_Majesty last updated on 30/Aug/20

$${f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{2}{x}}{\pi} \\$$$${f}_{\mathrm{1}} '\left({x}\right)=\frac{\mathrm{2}}{\pi} \\$$$${f}_{\mathrm{2}} \left({x}\right)={sinx} \\$$$${f}_{\mathrm{2}} '\left({x}\right)={cosx} \\$$$${f}_{\mathrm{3}} \left({x}\right)={x} \\$$$${f}_{\mathrm{3}} '\left({x}\right)=\mathrm{1} \\$$$$\\$$$$\mathrm{0}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}: \\$$$${f}_{\mathrm{1}} \left(\mathrm{0}\right)={f}_{\mathrm{2}} \left(\mathrm{0}\right)={f}_{\mathrm{3}} \left(\mathrm{0}\right)=\mathrm{0} \\$$$$\wedge \\$$$${f}_{\mathrm{1}} '\left({x}\right)\leqslant{f}_{\mathrm{2}} '\left({x}\right)\leqslant{f}_{\mathrm{3}} '\left({x}\right) \\$$$$\Rightarrow \\$$$${f}_{\mathrm{1}} \left({x}\right)\:{stays}\:\leqslant{f}_{\mathrm{2}} \left({x}\right) \\$$$${f}_{\mathrm{2}} \left({x}\right)\:{stays}\:\leqslant\:{f}_{\mathrm{3}} \left({x}\right) \\$$$${within}\:{the}\:{given}\:{interval} \\$$

Commented by ZiYangLee last updated on 31/Aug/20

$$\mathrm{Thanks}! \\$$