Question Number 174587 by bekzodjumayev last updated on 05/Aug/22
Commented by kaivan.ahmadi last updated on 05/Aug/22
![f(−x)=−f(x) ⇒f is an odd function ⇒f′(x)=f′(−x)⇒f′ is an even function ⇒fof(−x)=f(−f(x))=−fof(x) ⇒fof is an odd function ⇒[fof]′ is an even function ⇒(fof)′(−3)=(fof)′(3) ⇒(fof)′(3)−(fof)′(−3)=0](Q174608.png)
$${f}\left(−{x}\right)=−{f}\left({x}\right) \\ $$$$\Rightarrow{f}\:{is}\:{an}\:{odd}\:{function} \\ $$$$\Rightarrow{f}'\left({x}\right)={f}'\left(−{x}\right)\Rightarrow{f}'\:{is}\:{an}\:{even}\:{function} \\ $$$$\Rightarrow{fof}\left(−{x}\right)={f}\left(−{f}\left({x}\right)\right)=−{fof}\left({x}\right) \\ $$$$\Rightarrow{fof}\:{is}\:{an}\:{odd}\:{function} \\ $$$$\Rightarrow\left[{fof}\right]'\:{is}\:{an}\:{even}\:{function} \\ $$$$\Rightarrow\left({fof}\right)'\left(−\mathrm{3}\right)=\left({fof}\right)'\left(\mathrm{3}\right) \\ $$$$\Rightarrow\left({fof}\right)'\left(\mathrm{3}\right)−\left({fof}\right)'\left(−\mathrm{3}\right)=\mathrm{0} \\ $$$$ \\ $$
Commented by bekzodjumayev last updated on 05/Aug/22
$${Thank}\:{you} \\ $$