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HMMT 二月 2001 · CALC 赛 · 第 2 题

HMMT February 2001 — CALC Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. The polynomial 3 x − 250 x +735 x is interesting because it has the maximum possiblenumber of relative extrema and points of inflection at integer lattice points for a quinticpolynomial. What is the sum of the x -coordinates of these points?
解析

英文解析

  1. The polynomial 3 x − 250 x +735 x is interesting because it has the maximum possiblenumber of relative extrema and points of inflection at integer lattice points for a quinticpolynomial. What is the sum of the x -coordinates of these points?
    4 2
    Solution: The first derivative is 15 x − 750 x + 735, whose roots (which give the relativeextrema) sum to 750 / 15 = 50. The second derivative is 60 x − 1500 x , whose roots (which 3
    give the points of inflection) sum to 1500 / 60 = 25, for a grand total of 75 .