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Question Number 4904 by love math last updated on 20/Mar/16

Determine the smallest natural  value of n, so the function   y= 5x sin 5nx will be even.

$${Determine}\:{the}\:{smallest}\:{natural} \\ $$$${value}\:{of}\:{n},\:{so}\:{the}\:{function}\: \\ $$$${y}=\:\mathrm{5}{x}\:{sin}\:\mathrm{5}{nx}\:{will}\:{be}\:{even}. \\ $$

Commented by Yozzii last updated on 20/Mar/16

y(−x)=5(−x)sin(−5nx)  y(−x)=−5x{−sin(5nx)}  y(−x)=5xsin5nx=y(x)  y(x)=y(−x) for all n,x∈R.  ⇒ min(n)=1∈N.  If f(u)&h(u) are odd functions  then for g(u)=f(u)h(u), g(u) is  even. h=5x is odd and f=sin5nx  is odd⇒y=h×f=5xsin5nx is even  for all n,x∈R.

$${y}\left(−{x}\right)=\mathrm{5}\left(−{x}\right){sin}\left(−\mathrm{5}{nx}\right) \\ $$$${y}\left(−{x}\right)=−\mathrm{5}{x}\left\{−{sin}\left(\mathrm{5}{nx}\right)\right\} \\ $$$${y}\left(−{x}\right)=\mathrm{5}{xsin}\mathrm{5}{nx}={y}\left({x}\right) \\ $$$${y}\left({x}\right)={y}\left(−{x}\right)\:{for}\:{all}\:{n},{x}\in\mathbb{R}. \\ $$$$\Rightarrow\:{min}\left({n}\right)=\mathrm{1}\in\mathbb{N}. \\ $$$${If}\:{f}\left({u}\right)\&{h}\left({u}\right)\:{are}\:{odd}\:{functions} \\ $$$${then}\:{for}\:{g}\left({u}\right)={f}\left({u}\right){h}\left({u}\right),\:{g}\left({u}\right)\:{is} \\ $$$${even}.\:{h}=\mathrm{5}{x}\:{is}\:{odd}\:{and}\:{f}={sin}\mathrm{5}{nx} \\ $$$${is}\:{odd}\Rightarrow{y}={h}×{f}=\mathrm{5}{xsin}\mathrm{5}{nx}\:{is}\:{even} \\ $$$${for}\:{all}\:{n},{x}\in\mathbb{R}. \\ $$$$ \\ $$

Commented by Yozzii last updated on 20/Mar/16

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