Show that the curve y=2sin2x -4x is never increasing

Question

I am confused about how to differentiate 2sin2x or how to show that it is less than 0 and I have been trying to figure this out by various websites but it will not give me a clear answer. I would be very grateful for any guidance on this. 

Answer 1

It is sufficient to show that the derivative is never positive over the entire domain of the function.  Use the chain rule.

 

y'(x) = 4cos(2x) - 4

 

The first term on the right can never exceed 4, so y'(x) can never exceed 0, and hence y(x) is never increasing.

Answer 2

y'=2cos(2x)(2x)'-4=4(cos2x-1)≤0 , since |cos(2x)|≤1.

                            what we do here is we replace cos(2x) with 1, the        

                             largest possible value of cos(x) [cos(2x)]