Question Number 4904 by love math last updated on 20/Mar/16
$${Determine}\:{the}\:{smallest}\:{natural} \\ $$$${value}\:{of}\:{n},\:{so}\:{the}\:{function}\: \\ $$$${y}=\:\mathrm{5}{x}\:{sin}\:\mathrm{5}{nx}\:{will}\:{be}\:{even}. \\ $$
Commented by Yozzii last updated on 20/Mar/16
$${y}\left(−{x}\right)=\mathrm{5}\left(−{x}\right){sin}\left(−\mathrm{5}{nx}\right) \\ $$$${y}\left(−{x}\right)=−\mathrm{5}{x}\left\{−{sin}\left(\mathrm{5}{nx}\right)\right\} \\ $$$${y}\left(−{x}\right)=\mathrm{5}{xsin}\mathrm{5}{nx}={y}\left({x}\right) \\ $$$${y}\left({x}\right)={y}\left(−{x}\right)\:{for}\:{all}\:{n},{x}\in\mathbb{R}. \\ $$$$\Rightarrow\:{min}\left({n}\right)=\mathrm{1}\in\mathbb{N}. \\ $$$${If}\:{f}\left({u}\right)\&{h}\left({u}\right)\:{are}\:{odd}\:{functions} \\ $$$${then}\:{for}\:{g}\left({u}\right)={f}\left({u}\right){h}\left({u}\right),\:{g}\left({u}\right)\:{is} \\ $$$${even}.\:{h}=\mathrm{5}{x}\:{is}\:{odd}\:{and}\:{f}={sin}\mathrm{5}{nx} \\ $$$${is}\:{odd}\Rightarrow{y}={h}×{f}=\mathrm{5}{xsin}\mathrm{5}{nx}\:{is}\:{even} \\ $$$${for}\:{all}\:{n},{x}\in\mathbb{R}. \\ $$$$ \\ $$
Commented by Yozzii last updated on 20/Mar/16
