Problem:
Which of the following describes the largest subset of values of y within the closed interval [0,Ο] for which
sin(x+y)β€sin(x)+sin(y)
for every x between 0 and Ο, inclusive?
Answer Choices:
A. y=0
B. 0β€yβ€4Οβ
C. 0β€yβ€2Οβ
D. 0β€yβ€43Οβ
E. 0β€yβ€Ο
Solution:
If 0β€xβ€Ο and 0β€yβ€Ο, then sin(x)β₯0, sin(y)β₯0,cos(x)β€1, and cos(y)β€1. Therefore
sin(x+y)=sin(x)β
cos(y)+cos(x)β
sin(y)β€sin(x)+sin(y)
The given inequality holds for all y such that 0β€yβ€Οβ.
The problems on this page are the property of the MAA's American Mathematics Competitions