HMMT 二月 2001 · 冲刺赛 · 第 13 题

英文解析

13. [5] Find the number of real zeros of x − x − x + 2.
    3 2 ′ 2
    Solution: Let f ( x ) = x − x − x + 2, so f ( x ) = 3 x − 2 x − 1. The slope is zero when
    ( )
    1 1
    3 x − 2 x − 1 = 0, where x = − and x = 1. Now f > 0 and f (1) > 0, so there are no 2
    3 3
    zeros between x = − and x = 1. Since lim f ( x ) > 0, there are no zeros for x > 1. Since 1
    x → + ∞3
    lim f ( x ) < 0, there is one zero for x < − , for a total of 1 zero.1
    x →−∞3 1