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Mathematics
General Math
MHB Math Problem of the Week
Math POTW for Secondary and High School Students
Maximize (x1+x2)(x1+x3)x4 for Quartic Equation with Real Roots in [1/2,2]
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[QUOTE="anuttarasammyak, post: 6803351, member: 674023"] I have not solved the problem. The following is some observation. 1 It seems that the quantity, say ##I##, is maximum when large order of the solutions are [tex]x_4 \leq x_i \leq x_1 [/tex] and it is minimum, the reciprocal of the maximum, when [tex]x_1 \leq x_i \leq x_4 [/tex] 2 Looking at the coefficients we get [tex]a=x_1+x_2+x_3+x_4=\frac{d}{x_1}+\frac{d}{x_2}+\frac{d}{x_3}+\frac{d}{x_4}...(1)[/tex] [tex]d=x_1x_2x_3x_4...(2)[/tex] [tex]b=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4[/tex] Thus for an easy case of d=1, in the maximum case [tex]x_4=\beta^{-1},\ \ \{x_3,x_2\}=\{\alpha^{-1},\alpha\}, x_1=\beta[/tex] where [tex]1 \leq \alpha \leq \beta \leq 2[/tex] then [tex]I=1[/tex] The minimum is also 1, so I=1 for any case. (1) is written as [tex](x_1+x_4)(x_2x_3-1)+(x_1x_4-1)(x_2+x_3)=0[/tex] [tex](x_2+A)(x_3+A)=1+A^2[/tex] where [tex]A=\frac{x_1x_4-1}{x_1+x_4}[/tex] Here we know that if there is a pair of reciprocal in the roots, the remaining pair is also reciprocal and d=1,I=1. [/QUOTE]
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Forums
Mathematics
General Math
MHB Math Problem of the Week
Math POTW for Secondary and High School Students
Maximize (x1+x2)(x1+x3)x4 for Quartic Equation with Real Roots in [1/2,2]
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