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Minimum Value of Fraction

Source: 2002 China Second Round Olympiad

August 30, 2014

algebra unsolvedalgebra

Problem Statement

There are real numbers

a,b

and

c

and a positive number

x_1, x_2

such that

f(x)=x^3+ax^2+bx+c

has three real roots

x_1, x_2

and

x_3

satisfying

(1) x_2-x_1=\lambda

(2) x_3>\frac{1}{2}(x_1+x_2)

. Find the maximum value of

\frac{2a^3+27c-9ab}{\lambda^3}

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