Question Number 169497 by Huy last updated on 01/May/22 | ||
$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$ $$\mathrm{a}{f}\left(\mathrm{b}\right)+\mathrm{b}{f}\left(\mathrm{a}\right)=\left(\mathrm{a}+\mathrm{b}\right){f}\left(\sqrt{\mathrm{ab}}\right)\:\forall\mathrm{ab}>\mathrm{0} \\ $$ | ||
Answered by aleks041103 last updated on 02/May/22 | ||
$${let}\:{a}={x},\:{b}={x}+\mathrm{2}{dx} \\ $$ $$\sqrt{{ab}}=\sqrt{{x}\left({x}+\mathrm{2}{dx}\right)}={x}\left(\mathrm{1}+\mathrm{2}\frac{{dx}}{{x}}\right)^{\mathrm{1}/\mathrm{2}} ={x}+{dx} \\ $$ $${xf}\left({x}+\mathrm{2}{dx}\right)+\left({x}+\mathrm{2}{dx}\right){f}\left({x}\right)=\mathrm{2}\left({x}+{dx}\right){f}\left({x}+{dx}\right) \\ $$ $${f}\left({x}+{dx}\right)={f}+{f}\:'{dx}+\frac{{f}''}{\mathrm{2}}{dx}^{\mathrm{2}} \\ $$ $${f}\left({x}+\mathrm{2}{dx}\right)={f}+\mathrm{2}{f}\:'{dx}+\mathrm{2}{f}\:''{dx}^{\mathrm{2}} \\ $$ $${x}\left({f}+\mathrm{2}{f}\:'{dx}+\mathrm{2}{f}\:''{dx}^{\mathrm{2}} \right)+{xf}+\mathrm{2}{fdx}= \\ $$ $$=\mathrm{2}\left({x}+{dx}\right)\left({f}+{f}\:'{dx}+\frac{{f}''}{\mathrm{2}}{dx}^{\mathrm{2}} \right) \\ $$ $${xf}+{xf}'{dx}+{fdx}+{xf}''{dx}^{\mathrm{2}} = \\ $$ $$={xf}+{xf}'{dx}+{fdx}+\frac{\mathrm{1}}{\mathrm{2}}{xf}''{dx}^{\mathrm{2}} +{f}'{dx}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{f}''{dx}^{\mathrm{3}} \\ $$ $$\Rightarrow{xf}''{dx}^{\mathrm{2}} =\left(\frac{\mathrm{1}}{\mathrm{2}}{xf}''+{f}'\right){dx}^{\mathrm{2}} \\ $$ $$\frac{\mathrm{1}}{\mathrm{2}}{xf}''={f}' \\ $$ $${let}\:{f}'={g} \\ $$ $$\Rightarrow\frac{{dg}}{{dx}}=\frac{\mathrm{2}{g}}{{x}} \\ $$ $$\Rightarrow\frac{{dg}}{{g}}=\frac{\mathrm{2}{dx}}{{x}} \\ $$ $$\Rightarrow{d}\left({ln}\left({g}\right)−\mathrm{2}{ln}\left({x}\right)\right)=\mathrm{0} \\ $$ $$\Rightarrow{g}=\mathrm{3}{cx}^{\mathrm{2}} \\ $$ $$\Rightarrow{f}={cx}^{\mathrm{3}} +{s} \\ $$ $$\Rightarrow{a}\left({cb}^{\mathrm{3}} +{s}\right)+{b}\left({ca}^{\mathrm{3}} +{s}\right)=\left({a}+{b}\right)\left({c}\left({ab}\right)^{\mathrm{3}/\mathrm{2}} +{s}\right) \\ $$ $$\Rightarrow\left({a}+{b}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)=\left({a}+{b}\right)\left({ab}\right)^{\mathrm{3}/\mathrm{2}} \\ $$ $$\Rightarrow\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{2}} ={a}^{\mathrm{3}} {b}^{\mathrm{3}} \\ $$ $${which}\:{is}\:{generally}\:{not}\:{true}! \\ $$ $${it}\:{would}\:{be}\:{iff}\:{c}=\mathrm{0} \\ $$ $$\Rightarrow{f}\left({x}\right)={const}.\:{is}\:{the}\:{only}\:{solution}. \\ $$ $$ \\ $$ $${Note}! \\ $$ $${This}\:{applies}\:{only}\:{for}\:{continuous}\:{functions}. \\ $$ | ||