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-val

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

$$\mathrm{A}\mathrm{high}\mathrm{school}\mathrm{question}\mathrm{from}\mathrm{Japan}$$$$$$$$\mathrm{Let}\mathrm{P}\left(x\right)\mathrm{be}\mathrm{a}\mathrm{real}\mathrm{polynomial}\mathrm{of}\mathrm{degree}4$$$$\mathrm{and}{\mathrm{P}}^{\left(4\right)}\left(0\right)=72.$$$$\mathrm{If}\mathrm{there}\mathrm{exists}t\in \mathbb{R}\mathrm{and}m,n\in \mathbb{R}\mathrm{such}\mathrm{that}$$$${\mathrm{P}}^{″}\left(t\right)=0\mathrm{and}{\mathrm{P}}^{\prime }\left(t\right)={\mathrm{P}}^{\prime }(t+m)={\int }_t^{t+m}{\mathrm{P}}^{\prime }\left(x\right)dx=n$$$$\Rightarrow \mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}m\mathrm{and}n.$$

Commented by Her_Majesty last updated on 21/Aug/20

$$P\left(x\right)=3x^4+{bx}^3+{cx}^2+dx+e$$$$wehave7unknown(b,c,d,e,m,n,t)but$$$$only3equations\Rightarrow wecannotfindvalues$$$$ofanyoftheunknownbutofcoursewecan$$$$findi.e.m,n,tdependentonb,c,d,e$$

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