To determine where the function f(x) = 4 cos(2x) is decreasing, we first need to find the derivative of the function. The key to understanding the behavior of the function lies in its derivative.
The derivative of f(x) is given by:
f'(x) = -8 sin(2x)
To find where the function is decreasing, we need to set the derivative less than zero:
-8 sin(2x) < 0
This simplifies to:
sin(2x) > 0
The sine function is positive in the intervals where the angle is in the first and second quadrants. For sin(θ) > 0, we have:
0 < 2x < π or π < 2x < 2π
Dividing these inequalities by 2 gives us the intervals:
0 < x < π/2 and π/2 < x < π
Thus, the intervals on which the function f(x) = 4 cos(2x) is decreasing are:
- nπ < x < nπ + π/2
- nπ + π/2 < x < (n+1)π
where n is any integer. You can check specific intervals based on this condition to see whether the function is indeed decreasing.