Calculus of a Single Variable, Chapter 3, 3.3, Section 3.3, Problem 42

This function is continuous and differentiable on the given interval. It is increasing when f'(x)>0 and decreasing when f'(x)<0. Let's find f'(x).
f(x) = sin(x)*cos(x) + 5 = (1/2)*sin(2x) + 5.
f'(x) = cos(2x).
It is zero at 2x=pi/2+k*pi , or x=pi/4+(k*pi)/2 . There are four such points on (0, 2pi) : pi/4 , (3pi)/4 , (5pi)/4 and (7pi)/4 .
f'(x) is positive on (0, pi/4) , ((3pi)/4 , (5pi)/4 ) and ((7pi)/4, 2pi) , f is increasing on these intervals. f'(x) is negative on (pi/4, (3pi)/4) and ((5pi)/4, (7pi)/4) , f is decreasing on these intervals.
Therefore the relative extrema are pi/4 (maximum), (3pi)/4 (minimum), (5pi)/4 (maximum) and (7pi)/4 (minimum).