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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="bob012345, post: 6801450, member: 575631"] Here is my answer. [SPOILER] My answer is 1.5625 Consider the expression as a function; $$f(x_1,x_2,x_3,x_4) = \dfrac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}$$ we can take the partial derivatives to get maximize and minimum conditions. $$ \frac{\partial f}{\partial x_1} = \frac{x_4(x_1^2 - x_2x_3)}{x_1^2(x_4 + x_2)(x_4 + x_3)} = 0$$ $$ \frac{\partial f}{\partial x_2} = \frac{x_4(x_1 + x_3)(-x_1 + x_4)}{x_1 (x_4 + x_3)(x_4 + x_2)^2} = 0$$ $$ \frac{\partial f}{\partial x_3} = \frac{x_4(x_1 + x_2)(-x_1 + x_4)}{x_1 (x_4 + x_2)(x_4 + x_3)^2} = 0$$ $$ \frac{\partial f}{\partial x_4} = \frac{(x_1 + x_2)(x_1 + x_3)(-x_4^2 + x_2x_3)}{x_1(x_4 + x_2)(x_4 + x_3)^2} = 0$$ Since all the roots are real and positive we can disregard relations that involve one root being the negative of the other leaving the following ##x_1 = x_4##, ##x_1 = \sqrt{x_2x_3}## and ##x_4 = \sqrt{x_2x_3}## if ##x_1=x_4## then ##f(x_1,x_2,x_3,x_4)= 1## for all ##x_2,x_3## There are many solutions to the second and third conditions but they suggest three roots are equal. Given that if we set ##x_1=x_2=x_3=\frac{1}{2}## and ##x_4=2## or vice versa we get a minimum of 0.64. If we set ##x_2=x_3=x_4=\frac{1}{2}## and ##x_1=2## or vice versa, we get a maximum of 1.5625. I also explored intermediate values all of which were lower. If we put the condition for ##x_4## back into the original equation we get $$f = \frac{(x_1 + x_4)^2}{4x_1x_4}$$ which maximizes to 1.5625 when ##x_1=\frac{1}{2}## and ##x_4=2## or vice versa. The minimum is one. Also, if we put in the condition for ##x_1## back into the original equation we get $$f = \frac{4x_1x_4}{(x_1 + x_4)^2}$$ which minimizes to 0.64 when ##x_1=\frac{1}{2}## and ##x_4=2## or vice versa. The maximum is one. [\SPOILER][/SPOILER] [/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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