Science:Math Exam Resources/Courses/MATH100/December 2010/Question 08/Solution 1
< Science:Math Exam Resources | Courses/MATH100 | December 2010 | Question 08(Redirected from Science:Math Exam Resources/Courses/MATH100/December 2010/Question 8/Solution 1)
Notice that the function is periodic since the equator is shaped like a circle. Mathematically this in particular means
We exploit this fact by looking at
and
At this point we want to distinguish two cases:
Case 1. f(0) = 0. Then
and hence , which is what we wanted to show.
Case 2. f(0) ≠ 0. Then we have that one of and is positive and one is negative. As T is continuous, we have that f is continuous and hence we may invoke the intermediate value theorem to see that there exists a point such that . Thus, we have and this completes the proof.