Find the critical numbers of U=U(t)

Question

Determine the intervals on which U is increasing and on which it is decreasing.
Find the local extrema of U.
U=U(t)=(5cos(2t/pi))/(2+sin2t/pi)+6

Answer 1

recall that critical numbers are where U'=0 or is undefined.
U is increasing where U' > 0
Now, since

U'(t) = -10/pi * (2sin(2t/pi)+1)/(sin(2t/pi)+2)^2

U'=0 when 2sin(2t/pi)+1 = 0
and since the denominator is always positive, U' is never undefined.
And, of course, U' > 0 when 2sin(2t/pi)+1 < 0