(C3 question). Find an expression for all stationary points on the curve y=sin(x)cos(x). How many such points are there and why?
Stationary points are points at which the gradient of the curve is zero. The gradient is given by dy/dx so we start by computing this using product rule to give us dy/dx=-sin2x+cos2x. Notice that this is the double angle formula for cos(2x) so we see that dy/dx=cos(2x). Now if we set the gradient equal to zero then dy/dx=cos(2x)=0, and we know that the cosine function is zero at the points (pi/2+kpi) where k is an integer. Hence we may set 2x=pi/2+kpi which gives us our solution x=pi/4+k*pi/2.Notice that this is true for any integer k so there are infinitely many stationary points. This is the case because sine and cosine are periodic functions so their product, y=sin(x)cos(x) is also periodic.
