2018 AMC 12A Problems/Problem 9
Problem
Which of the following describes the largest subset of values of within the closed interval for which for every between and , inclusive?
Solution 1
On the interval sine is nonnegative; thus for all and equality only occurs when , which is cosine's maximum value. The answer is . (CantonMathGuy)
Solution 2
Expanding, Let , . We have that Comparing coefficients of and gives a clear solution: both and are less than or equal to one, so the coefficients of and on the left are less than on the right. Since , that means that this equality is always satisfied over this interval, or .
Solution 3
If we plug in , we can see that . Note that since is always nonnegative, is always nonpositive. So, the inequality holds true when . The only interval that contains in the answer choices is .
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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