Question Number 109128 by ZiYangLee last updated on 21/Aug/20
$$\mathrm{A}\:\mathrm{high}\:\mathrm{school}\:\mathrm{question}\:\mathrm{from}\:\mathrm{Japan} \\ $$$$ \\ $$$$\mathrm{Let}\:\mathrm{P}\left({x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{4}\: \\ $$$$\mathrm{and}\:\mathrm{P}^{\left(\mathrm{4}\right)} \left(\mathrm{0}\right)=\mathrm{72}. \\ $$$$\mathrm{If}\:\mathrm{there}\:\mathrm{exists}\:{t}\in\mathbb{R}\:\mathrm{and}\:{m},{n}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{P}''\left({t}\right)=\mathrm{0}\:\mathrm{and}\:\mathrm{P}'\left({t}\right)=\mathrm{P}'\left({t}+{m}\right)=\int_{{t}} ^{\:{t}+{m}} \mathrm{P}'\left({x}\right){dx}={n} \\ $$$$\Rightarrow\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:\mathrm{and}\:{n}. \\ $$
Commented by Her_Majesty last updated on 21/Aug/20
$${P}\left({x}\right)=\mathrm{3}{x}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e} \\ $$$${we}\:{have}\:\mathrm{7}\:{unknown}\:\left({b},\:{c},\:{d},\:{e},\:{m},\:{n},\:{t}\right)\:{but} \\ $$$${only}\:\mathrm{3}\:{equations}\:\Rightarrow\:{we}\:{cannot}\:{find}\:{values} \\ $$$${of}\:{any}\:{of}\:{the}\:{unknown}\:{but}\:{of}\:{course}\:{we}\:{can} \\ $$$${find}\:{i}.{e}.\:{m},\:{n},\:{t}\:{dependent}\:{on}\:{b},\:{c},\:{d},\:{e} \\ $$