2017 AMC 12A Problems/Problem 15

Problem

Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x) = 0$ lie?

$\textbf{(A)}\ (0,1)  \qquad \textbf{(B)}\ (1, 2) \qquad\textbf{(C)}\ (2, 3) \qquad\textbf{(D)}\ (3, 4) \qquad\textbf{(E)}\ (4,5)$

Solution

We must first get an idea of what $f(x)$ looks like:

Between 0 and 1, $f(x)$ starts at $2$ and increases; clearly there is no zero here.

Between 1 and $\frac{\pi}{2}$, $f(x)$ starts at a positive number and increases to $\infty$; there is no zero here either.

Between $\frac{\pi}{2}$ and 3, $f(x)$ starts at $-\infty$ and increases to some negative number; there is no zero here either.

Between 3 and $\pi$, $f(x)$ starts at some negative number and increases to -2; there is no zero here either.

Between $\pi$ and $\pi+\frac{\pi}{4} < 4$, $f(x)$ starts at -2 and increases to $-\frac{\sqrt2}{2} + 2\left(-\frac{\sqrt2}{2}\right) + 3\left(1\right)=3\left(1-\frac{\sqrt2}{2}\right)>0$. There is a zero here by the Intermediate Value Theorem. Therefore, the answer is $\boxed{\textbf{(D)}}$.

Solution 2

If you quickly take a moment to sketch the graphs of the three functions, you will see that between 0 and pi/2 everything is positive, while the positive number created by the sin does not outweigh the negative by the cos and tan function. Upon further examination, it is clear that the positive the tan function creates will balance the other two functions, and thus the first solution is a little bit after pi, which is around 3.14. Hence the answer is $\boxed{\textbf{(D)}}$.

-note: solution makes more sense if you just sketch it

Solution by roadchicken~

(not original author) Here is the graph: [asy] Label f;  f.p=fontsize(6);  xaxis(-5,5,Ticks(f, 1.0));  yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return sin(x); } draw(graph(f, -5,5)); real g(real x) { return 2*cos(x); } draw(graph(g, -5,5)); real h(real x) { return 3*tan(x); } draw(graph(h, -1.2,1.2)); draw(graph(h, 1.94, 4.34)); draw(graph(h, -4.34, -1.94)); [/asy]

See Also

2017 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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
All AMC 12 Problems and Solutions

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