f(x)=sin(2x) for 0<x<pi, find the values of x for which f is a decreasing function

A decreasing function means that its instantaneous slope (rate of change) is negative and the function tends to smaller values of y.There are two approaches to this. One is graphical, as sin(x) is a recognisable function and simply by picturing it we can give an answer: (pi/2,3pi/2). This makes sense because sin(x) reaches its maximum of 1 at pi/2, and from that point on decreases until its minimum value of -1. Since sin(2x) is simply a horizontal expansion of sin(x) by a factor of 1/2, we can correct our domain for the values of x accordingly: (pi/4, 3pi/4). The other approach is to use derivatives. We know d/dx(sin(2x))=2cos2x, hence we have: f'(x)=2cos(2x). We will now find the points where f'(x) is 0, meaning that the "slope" is flat and the function has a turning point. In the domain 0<x<pi, our two turning points will be at pi/4 and 3pi/4. We can now draw a sign diagram to see where the slope of f'(x) is negative, and reach the answer (pi/4, 3pi/4).