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1999 AHSME Problems/Problem 18

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Problem

How many zeros does $f(x) = \cos(\log x)$ have on the interval $0 < x < 1$ ?

$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } \text{in...$

Solution

For $0 < x < 1$ we have $-\infty < \log x < 0$ , and the logarithm is a strictly increasing function on this interval.

$\cos(t)$ is zero for all $t$ of the form $\frac{\pi}2 + k\pi$ , where $k\in\mathbb{Z}$ . There are $\boxed{\text{infinitely\ many}}$ such $t$ in $(-\infty,0)$ .

Here's the graph of the function on $(0,1)$ :

$import graph;size(250,200,IgnoreAspect);real f(real t) {return cos(log(t));}draw(graph(f,0.01,1),red,"$\cos(\log(x))$&qu...$

As we go closer to $0$ , the function will more and more wildly oscilate between $-1$ and $1$ . This is how it looks like at $(0.0001,0.02)$ .

$import graph;size(250,200,IgnoreAspect);real f(real t) {return cos(log(t));}draw(graph(f,0.0001,0.02),red,"$\cos(\log(x)...$

And one more zoom, at $(0.000001,0.0005)$ .

$import graph;size(250,200,IgnoreAspect);real f(real t) {return cos(log(t));}draw(graph(f,0.000001,0.0005),red,"$\cos(\lo...$

See also

1999 AHSME (Problems)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30

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