1999 AMC 12 Problem 18

Attempt Problem 18 of the 1999 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

How many zeros does

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

have on the interval

0 \lt x \lt 1?

0

1

2

10

infinitely many

x

ranges over

(0, 1),

what values does

\log x

take?

\log x

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

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.

Problem 18 in other years