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Category: Integration

Determine-all-functions-f-0-1-such-that-x-0-1-f-x-f-x-f-0-f-1-

Question Number 88003 by Ar Brandon last updated on 07/Apr/20 $${Determine}\:{all}\:{functions}\:{f}\left[\mathrm{0},\mathrm{1}\right]\rightarrow\Omega \\ $$$${such}\:{that}\:\forall{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{f}\:'\left({x}\right)+{f}\left({x}\right)={f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right) \\ $$ Commented by mr W last updated on 07/Apr/20 $${f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)={k} \\…

Question-87969

Question Number 87969 by M±th+et£s last updated on 07/Apr/20 Answered by mind is power last updated on 07/Apr/20 $$=\underset{{k}\geqslant\mathrm{1}} {\sum}\int_{{k}} ^{\frac{\mathrm{2}{k}+\mathrm{1}}{\mathrm{2}}} \frac{\sqrt{\left.{x}−\lfloor{x}\right]}}{\left[\mathrm{2}{x}\right]^{\mathrm{2}} }{d}\underset{={S}} {{x}}+\underset{{k}\geqslant\mathrm{1}} {\sum}\int_{\frac{\mathrm{2}{k}+\mathrm{1}}{\mathrm{2}}}…