To find the absolute maximum and absolute minimum values of a function over a bounded interval, we need to evaluate the function at critical points and endpoints, then determine the largest and smallest function values.
1. First, let's find out the critical points for the function f(x) = 2cosx + cos(2x). A critical point occurs where the function's derivative is zero or undefined.
2. The derivative, f'(x), of the function is -2sinx - 2sin(2x). Setting this equal to zero gives us a way to find the critical points: -2sinx - 2sin(2x) = 0.
3. Solve this equation to get the critical points. Unfortunately, this equation is not one that we can solve easily using elementary algebra, so we would typically resort to numerical solution techniques for finding the approximate values of x.
4. Once we have found the critical points, we evaluate the function f at these points, as well as at the endpoints of the interval, which are 0 and 2π.
5. In this case, we found one critical point x = 2.094, and the two endpoint values are x = 0 and x = 2π.
6. Evaluate the function at these three points to get f(2.094) = -1.500, f(0)= 3.000, and f(2π) = 3.000.
7. The largest of these values is the absolute maximum of the function over the given interval, and the smallest of these values is the absolute minimum.
So, the absolute maximum value of this function on the interval [0, 2π] is 3.000, and it occurs at x = 0 and x = 2π. The absolute minimum value is -1.500, and it occurs at x = 2.094.