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Question Number 109128 by ZiYangLee last updated on 21/Aug/20

A high school question from Japan Let P(x) be a real polynomial of degree 4 and P^((4)) (0)=72. If there exists t∈R and m,n∈R such that P′′(t)=0 and P′(t)=P′(t+m)=∫_t ^( t+m) P′(x)dx=n ⇒Find the value of m and n.

$$\mathrm{A}\:\mathrm{high}\:\mathrm{school}\:\mathrm{question}\:\mathrm{from}\:\mathrm{Japan} \\ $$$$ \\ $$$$\mathrm{Let}\:\mathrm{P}\left({x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{4}\: \\ $$$$\mathrm{and}\:\mathrm{P}^{\left(\mathrm{4}\right)} \left(\mathrm{0}\right)=\mathrm{72}. \\ $$$$\mathrm{If}\:\mathrm{there}\:\mathrm{exists}\:{t}\in\mathbb{R}\:\mathrm{and}\:{m},{n}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{P}''\left({t}\right)=\mathrm{0}\:\mathrm{and}\:\mathrm{P}'\left({t}\right)=\mathrm{P}'\left({t}+{m}\right)=\int_{{t}} ^{\:{t}+{m}} \mathrm{P}'\left({x}\right){dx}={n} \\ $$$$\Rightarrow\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:\mathrm{and}\:{n}. \\ $$

Commented by Her_Majesty last updated on 21/Aug/20

P(x)=3x^4 +bx^3 +cx^2 +dx+e we have 7 unknown (b, c, d, e, m, n, t) but only 3 equations ⇒ 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

$${P}\left({x}\right)=\mathrm{3}{x}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e} \\ $$$${we}\:{have}\:\mathrm{7}\:{unknown}\:\left({b},\:{c},\:{d},\:{e},\:{m},\:{n},\:{t}\right)\:{but} \\ $$$${only}\:\mathrm{3}\:{equations}\:\Rightarrow\:{we}\:{cannot}\:{find}\:{values} \\ $$$${of}\:{any}\:{of}\:{the}\:{unknown}\:{but}\:{of}\:{course}\:{we}\:{can} \\ $$$${find}\:{i}.{e}.\:{m},\:{n},\:{t}\:{dependent}\:{on}\:{b},\:{c},\:{d},\:{e} \\ $$

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