Question Number 143814 by mathdanisur last updated on 18/Jun/21
$$\forall{a};{b};{c}\in\mathbb{R}\:,\:{find}\:{all}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:,\:{such}\:{that} \\ $$$${f}\left({a}\right){f}\left({bc}\right)+\mathrm{9}\leqslant{f}\left({ab}\right)+\mathrm{5}{f}\left({ac}\right) \\ $$
Answered by Olaf_Thorendsen last updated on 18/Jun/21
$${f}\left({a}\right){f}\left({bc}\right)+\mathrm{9}\:\leqslant\:{f}\left({ab}\right)+\mathrm{5}{f}\left({ac}\right) \\ $$$$ \\ $$$$\mathrm{If}\:{a}\:=\:{b}\:=\:{c}\:=\:\mathrm{0}\:: \\ $$$${f}^{\mathrm{2}} \left(\mathrm{0}\right)+\mathrm{9}\:\leqslant\:\mathrm{6}{f}\left(\mathrm{0}\right) \\ $$$${f}^{\mathrm{2}} \left(\mathrm{0}\right)−\mathrm{6}{f}\left(\mathrm{0}\right)+\mathrm{9}\:\leqslant\:\mathrm{0} \\ $$$$\left({f}\left(\mathrm{0}\right)−\mathrm{3}\right)^{\mathrm{2}} \:\leqslant\:\mathrm{0}\:\Rightarrow\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{3} \\ $$$$ \\ $$$$\mathrm{If}\:{a}\:=\:{b}\:=\:{c}\:=\:\mathrm{1}\:: \\ $$$${f}\left(\mathrm{1}\right)\:=\:\mathrm{3} \\ $$$$ \\ $$$$\mathrm{If}\:{a}\:=\:\mathrm{0}\:\mathrm{and}\:\mathrm{c}=\:\mathrm{1}\:: \\ $$$${f}\left(\mathrm{0}\right){f}\left({b}\right)+\mathrm{9}\:\leqslant\:{f}\left(\mathrm{0}\right)+\mathrm{5}{f}\left(\mathrm{0}\right) \\ $$$$\mathrm{3}{f}\left({b}\right)+\mathrm{9}\:\leqslant\:\mathrm{6}{f}\left(\mathrm{0}\right)\:=\:\mathrm{18} \\ $$$${f}\left({b}\right)\:\leqslant\:\mathrm{3}\:\forall{b}\:\:\:\left(\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{If}\:{a}\:=\:\mathrm{1}\:\mathrm{and}\:{b}\:=\:\mathrm{0}\:: \\ $$$${f}\left(\mathrm{1}\right){f}\left(\mathrm{0}\right)+\mathrm{9}\:\leqslant\:{f}\left(\mathrm{0}\right)+\mathrm{5}{f}\left({c}\right) \\ $$$$\mathrm{15}\:\leqslant\:\mathrm{5}{f}\left({c}\right) \\ $$$${f}\left({c}\right)\:\geqslant\:\mathrm{3}\:\forall{c}\:\:\:\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\mathrm{and}\:\left(\mathrm{2}\right)\:\forall{x},\:{f}\left({x}\right)\:=\:\mathrm{3} \\ $$
Commented by mathdanisur last updated on 20/Jun/21
$${thankyou}\:{sir}\:{cool} \\ $$