Question Number 50161 by cesar.marval.larez@gmail.com last updated on 14/Dec/18
$${Find}\:{the}\:{function}\:{whose}\:{first}\: \\ $$$${derivative}\:{is}\:\mathrm{8}−\frac{\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }}\:{the}\:{initial}\: \\ $$$${conditions}\:{f}\left(\mathrm{8}\right)=−\mathrm{20} \\ $$
Answered by afachri last updated on 14/Dec/18
$${F}\left({x}\right)\:\:=\:\:\int\:{f}'\left({x}\right)\:\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\int\:\:\mathrm{8}\:−\:\mathrm{5}{x}^{−\frac{\mathrm{2}}{\mathrm{3}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{8}{x}\:\:−\:\:\left(\frac{\mathrm{5}}{\left(−\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{1}\right)}\right){x}^{−\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{1}} \:+\:\:{C} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{8}{x}\:−\:\:\mathrm{15}{x}\:^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:{C} \\ $$$$\mathrm{to}\:\mathrm{find}\:{C}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{F}\left(\mathrm{8}\right)\:\:=\:\:−\mathrm{20} \\ $$$$\mathrm{8}\left(\mathrm{8}\right)\:\:−\:\:\mathrm{15}\left(\mathrm{8}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{C}\:\:=\:\:−\mathrm{20} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{64}\:\:−\:\:\mathrm{30}\:\:+\:\:{C}\:\:=\:\:−\mathrm{20} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{C}\:\:=\:\:−\mathrm{54} \\ $$$$ \\ $$$$\mathrm{So}\:\::\:\:{F}\left({x}\right)\:\:=\:\:\mathrm{8}{x}\:\:−\:\:\mathrm{15}{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\:\:\mathrm{54} \\ $$$$ \\ $$$$ \\ $$
Commented by cesar.marval.larez@gmail.com last updated on 14/Dec/18
$${wow}\:{friend}\:{my}\:{respect}\:{i}\:{will}\:{the}\:{others} \\ $$
Commented by afachri last updated on 14/Dec/18
$$\mathrm{it}'\mathrm{s}\:\mathrm{been}\:\mathrm{our}\:\mathrm{pleasure} \\ $$