Question Number 149188 by mathdanisur last updated on 03/Aug/21
$${f}\left({x}\right)\:=\:{ax}^{\mathrm{99}} \:+\:{bx}^{\mathrm{77}} \:+\:{cx}^{\mathrm{55}} \:+\:\mathrm{975} \\ $$$${f}\left(−\mathrm{975}\right)\:=\:\mathrm{1000} \\ $$$${find}\:\:\:{f}\left(\mathrm{975}\right)\:=\:? \\ $$
Answered by mr W last updated on 03/Aug/21
$${g}\left({x}\right)={f}\left({x}\right)−\mathrm{975}={ax}^{\mathrm{99}} +{bx}^{\mathrm{77}} +{cx}^{\mathrm{55}} \\ $$$${g}\left(−{x}\right)=−{g}\left({x}\right)\:\Rightarrow{odd}\:{function} \\ $$$${g}\left(−\mathrm{975}\right)=−{g}\left(\mathrm{975}\right) \\ $$$${f}\left(−\mathrm{975}\right)−\mathrm{975}=−\left({f}\left(\mathrm{975}\right)−\mathrm{975}\right) \\ $$$${f}\left(−\mathrm{975}\right)=\mathrm{2}×\mathrm{975}−{f}\left(\mathrm{975}\right) \\ $$$${f}\left(−\mathrm{975}\right)=\mathrm{2}×\mathrm{975}−\mathrm{1000}=\mathrm{950} \\ $$
Commented by mathdanisur last updated on 03/Aug/21
$${Ser},\:{Thank}\:{You} \\ $$
Answered by liberty last updated on 03/Aug/21
$$\Rightarrow\mathrm{1000}=−\mathrm{a}\left(\mathrm{975}\right)^{\mathrm{99}} −\mathrm{b}\left(\mathrm{975}\right)^{\mathrm{77}} −\mathrm{c}\left(\mathrm{975}\right)^{\mathrm{55}} +\mathrm{975} \\ $$$$\Rightarrow−\mathrm{25}=\mathrm{a}\left(\mathrm{975}\right)^{\mathrm{99}} +\mathrm{b}\left(\mathrm{975}\right)^{\mathrm{77}} +\mathrm{c}\left(\mathrm{975}\right)^{\mathrm{55}} \\ $$$$\mathrm{f}\left(\mathrm{975}\right)=\mathrm{a}\left(\mathrm{975}\right)^{\mathrm{99}} +\mathrm{b}\left(\mathrm{975}\right)^{\mathrm{77}} +\mathrm{c}\left(\mathrm{975}\right)^{\mathrm{55}} +\mathrm{975} \\ $$$$=\:−\mathrm{25}+\mathrm{975}=\mathrm{950} \\ $$
Commented by mathdanisur last updated on 03/Aug/21
$${Ser},\:{Thank}\:{You}\: \\ $$