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f-R-R-g-R-R-f-xg-f-x-g-y-g-xy-f-x-g-y-d-fg-dx-




Question Number 821 by 123456 last updated on 17/Mar/15
f:R→R  g:R→R  f(xg)=f(x)g(y)  g(xy)=f(x)+g(y)  ((d(fg))/dx)=?
$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({xg}\right)={f}\left({x}\right){g}\left({y}\right) \\ $$$${g}\left({xy}\right)={f}\left({x}\right)+{g}\left({y}\right) \\ $$$$\frac{{d}\left({fg}\right)}{{dx}}=? \\ $$
Answered by prakash jain last updated on 18/Mar/15
g(1)=f(1)+g(1)⇒f(1)=0  f(xy)=f(x)g(y)⇒f(y)=f(1)g(y)=0  f(x)=0  g(xy)=g(y)  g(x)=g(1) ⇒g(x)=Constant  f(x)g(x)=0
$${g}\left(\mathrm{1}\right)={f}\left(\mathrm{1}\right)+{g}\left(\mathrm{1}\right)\Rightarrow{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$${f}\left({xy}\right)={f}\left({x}\right){g}\left({y}\right)\Rightarrow{f}\left({y}\right)={f}\left(\mathrm{1}\right){g}\left({y}\right)=\mathrm{0} \\ $$$${f}\left({x}\right)=\mathrm{0} \\ $$$${g}\left({xy}\right)={g}\left({y}\right) \\ $$$${g}\left({x}\right)={g}\left(\mathrm{1}\right)\:\Rightarrow{g}\left({x}\right)={C}\mathrm{onstant} \\ $$$${f}\left({x}\right){g}\left({x}\right)=\mathrm{0} \\ $$

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