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The minimum value of a function

Source: Indonesian MO (INAMO) 2009, Day 2, Problem 6

August 8, 2009
functioninequalities unsolvedinequalities

Problem Statement

Find the lowest possible values from the function f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009 for any real numbers x x.