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Largest Sum of 2 coeff. knowing relation between 2 roots
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freemind
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#1
Largest Sum of 2 coeff. knowing relation between 2 roots
Moldova National MO 2008, 12 Grade, Problem 1
Consider the equation x^4 - 4x^3 + 4x^2 + ax + b = 0, where a,b\in\mathbb{R}. Determine the largest value a + b can take, so that the given equation has two distinct positive roots x_1,x_2 so that x_1 + x_2 = 2x_1x_2.
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PostPosted: Sat Mar 01, 2008 4:32 pm
AYMANE
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#2
Write P as P(x)=(x^2+Ax+B)(x^2-2t+t) where t=x_1x_2>0
This gives:
A=2t-4;B=4t^2-9t+4;a=At-2Bt;b=Bt
\implies a+b=(A-B)t=-4t^3+11t^2=f(t)
f'(t)=-12t^2+22t=0
\implies t=0;t=\frac{22}{12} but f(0)=0 ;f(\frac{22}{12})=\frac{484}{33};f(+\infty)=-\infty
\implies Max(a+b)=\frac{484}{33}
this Max is realised for x_1=\frac{22+\sqrt{220}}{12}>0;x_2=\frac{22-\sqrt{220}}{12}>0
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PostPosted: Thu Mar 06, 2008 7:39 pm
G.Huvent
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#3
I think that A-B=-4t²+11t-8

PostPosted: Mon Mar 10, 2008 5:13 pm
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