HMMT 二月 2001 · 冲刺赛 · 第 13 题
HMMT February 2001 — Guts Round — Problem 13
题目详情
- [5] Find the number of real zeros of x − x − x + 2. 1
解析
- [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 2 3 x − 2 x − 1 = 0, where x = − and x = 1. Now f > 0 and f (1) > 0, so there are no 3 3 1 zeros between x = − and x = 1. Since lim f ( x ) > 0, there are no zeros for x > 1. Since 3 x → + ∞ 1 lim f ( x ) < 0, there is one zero for x < − , for a total of 1 zero. 3 x →−∞ 1