Question Number 128750 by bemath last updated on 10/Jan/21
$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\:\mathrm{satisfy}\:\mathrm{f}\left(−\mathrm{x}\right)=\mathrm{3f}\left(\mathrm{x}\right). \\ $$$$\mathrm{If}\:\int_{−\mathrm{1}} ^{\:\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{2}\:\mathrm{then}\:\int_{−\mathrm{2}} ^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=? \\ $$
Commented by mr W last updated on 10/Jan/21
$${if}\:{f}\left(−{x}\right)=\mathrm{3}{f}\left({x}\right)\:{is}\:{valid}\:{for}\:{any}\:{x}\in{R}, \\ $$$${then}\:{question}\:{is}\:{wrong},\:{since}\:{f}\left({x}\right)=\mathrm{0} \\ $$$${and}\:\int_{−\mathrm{1}} ^{\mathrm{2}} {f}\left({x}\right){dx}=\mathrm{0}\neq\mathrm{2}. \\ $$$$ \\ $$$${if}\:{f}\left(−{x}\right)=\mathrm{3}{f}\left({x}\right)\:{is}\:{valid}\:{for}\:{x}\in{R}^{+} , \\ $$$${then}\:{you}\:{can}'{t}\:{determine}\:\int_{−\mathrm{2}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$${from}\:{the}\:{single}\:{condition} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{2}} {f}\left({x}\right){dx}=\mathrm{2}. \\ $$$$ \\ $$$${so}\:{please}\:{recheck}\:{the}\:{question}! \\ $$
Answered by bramlexs22 last updated on 10/Jan/21
$$\mathrm{replace}\:\mathrm{x}\:\mathrm{by}\:−\mathrm{x}\:\mathrm{then}\:\mathrm{we}\:\mathrm{find} \\ $$$$\:\int_{−\mathrm{2}} ^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\int_{\mathrm{2}} ^{\:−\mathrm{1}} \mathrm{f}\left(−\mathrm{x}\right)\:\mathrm{d}\left(−\mathrm{x}\right) \\ $$$$\:=\:\int_{−\mathrm{1}} ^{\:\mathrm{2}} \mathrm{f}\left(−\mathrm{x}\right)\:\mathrm{dx}\:=\:\int_{−\mathrm{1}} ^{\:\mathrm{2}} \mathrm{3f}\left(\mathrm{x}\right)\:\mathrm{dx}\: \\ $$$$\:=\:\mathrm{3}\:×\mathrm{2}\:=\:\mathrm{6}\:. \\ $$
Commented by mr W last updated on 10/Jan/21
$${if}\:{f}\left(−{x}\right)=\mathrm{3}{f}\left({x}\right)\:{for}\:{x}\in{R},\:{then} \\ $$$${f}\left({x}\right)=\mathrm{3}{f}\left(−{x}\right),\:{and}\:{then} \\ $$$${f}\left({x}\right)=\mathrm{3}{f}\left(−{x}\right)=\mathrm{9}{f}\left({x}\right),\:{and}\:{then} \\ $$$${f}\left({x}\right)=\mathrm{0},\:{and}\:{then} \\ $$$$\:\int_{−\mathrm{2}} ^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{0},\:\:\int_{−\mathrm{1}} ^{\:\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{0}\neq\mathrm{2}. \\ $$
Commented by bramlexs22 last updated on 10/Jan/21
$$\mathrm{hahaha}....\mathrm{it}\:\mathrm{follows}\:\mathrm{that}\:\mathrm{question}\:\mathrm{wrong} \\ $$