found-something-others-have-found-before-which-I-thought-might-be-of-interest-especially-for-Sir-Tanmay-Chaudhury-take-any-polynome-of-degree-4-with-2-real-inflection-points-y-ax-4-bx-3-cx-2-dx

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Question Number 43461 by MJS last updated on 10/Sep/18

$$\mathrm{found}\mathrm{something}\left(\mathrm{others}\mathrm{have}\mathrm{found}\mathrm{before}\right)$$$$\mathrm{which}\mathrm{I}\mathrm{thought}\mathrm{might}\mathrm{be}\mathrm{of}\mathrm{interest},$$$$\mathrm{especially}\mathrm{for}\mathrm{Sir}\mathrm{Tanmay}\mathrm{Chaudhury}:$$$$\mathrm{take}\mathrm{any}\mathrm{polynome}\mathrm{of}\mathrm{degree}4\mathrm{with}2\mathrm{real}$$$$\mathrm{inflection}\mathrm{points}$$$$y={ax}^4+{bx}^3+{cx}^2+dx+e$$$$y^{″}=12{ax}^2+6bx+2c=0\mathrm{has}\mathrm{got}2\mathrm{real}\mathrm{solutions}$$$$x_1\mathrm{and}{x}_2$$$$\mathrm{the}\mathrm{line}\mathrm{connecting}\mathrm{the}\mathrm{inflection}\mathrm{points}$$$$\mathrm{intersects}\mathrm{the}\mathrm{curve}\mathrm{in}2\mathrm{more}\mathrm{points}$$$$P\mathrm{and}Q,\mathrm{their}x-\mathrm{values}\mathrm{are}p\mathrm{and}q$$$$\mathrm{let}p<x_1<x_2<q$$$$\Rightarrow \frac{x_2-x_1}{x_1-p}=\frac{x_2-x_1}{q-x_2}=\frac{1}{2}+\frac{\sqrt{5}}{2}\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{Golden}\mathrm{Ratio}$$

Commented by tanmay.chaudhury50@gmail.com last updated on 11/Sep/18

$$thankyousir\dots goldenratioiskeytoknowuniverse\dots$$

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