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Question Number 128424 by bramlexs22 last updated on 07/Jan/21
$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function}\:\mathrm{and} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{x}^{\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)−\mathrm{2f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\mathrm{where}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function}\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{5}\right)\:\mathrm{is}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{0}\:\:\left(\mathrm{b}\right)\:\frac{\mathrm{37}}{\mathrm{75}}\:\:\:\left(\mathrm{c}\right)\:\mathrm{4}\:\:\:\:\left(\mathrm{d}\right)\:\frac{\mathrm{51}}{\mathrm{77}} \\ $$
Answered by mr W last updated on 07/Jan/21
$${x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)={g}\left({x}\right)\:\:\:...\left({i}\right) \\ $$$$\left(−{x}\right)^{\mathrm{2}} {f}\left(−{x}\right)−\mathrm{2}{f}\left(−{x}\right)={g}\left(−{x}\right) \\ $$$${x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)=−{g}\left({x}\right)\:\:\:...\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right): \\ $$$$\mathrm{2}\left({x}^{\mathrm{2}} {f}\left({x}\right)−\mathrm{2}{f}\left({x}\right)\right)=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{2}\right){f}\left({x}\right)=\mathrm{0} \\ $$$$\left(\mathrm{5}^{\mathrm{2}} −\mathrm{2}\right){f}\left(\mathrm{5}\right)=\mathrm{0} \\ $$$$\Rightarrow{f}\left(\mathrm{5}\right)=\mathrm{0} \\ $$