1999 AMC 12 第 18 题

Question

函数

f(x) = \cos(\log x)

在区间

0 \lt x \lt 1

上有多少个零点？

How many zeros does

f(x) = \cos(\log x)

have on the interval

0 \lt x \lt 1?

0

1

2

10

无限多个

infinitely many

当

x

在

(0, 1)

中变化时，

\log x

取哪些值？

As

x

ranges over

(0, 1),

what values does

\log x

take?

\log x

覆盖所有负数，而余弦函数在负数中有无限多个零点。

\log x

covers all negative numbers, and cosine vanishes at infinitely many of them

当

x

在

(0, 1)

中变化时，

\log x

覆盖所有负实数。余弦函数在

\tfrac{\pi}{2} - n\pi

处为零，其中

n

取遍每个正整数，而这些值全都是负数，因此

f

有无限多个零点。

所以正确答案是 E。

As

x

ranges over

(0, 1),

\log x

ranges over all negative real numbers. The cosine function is zero at

\tfrac{\pi}{2} - n\pi

for every positive integer

n,

all of which are negative, so

f

has infinitely many zeros.

Thus, the correct answer is E.

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