Answer:
6π ≈ 18.8495559215
Explanation:
You want to know the absolute maximum of f(x) = 3x +sin(3x) on the interval [0, 2π].
Slope
The function is monotonically increasing on the interval, so has its maximum at the right end of the interval:
Fmax = f(2π) = 3(2π) +sin(3(2π)) = 6π +0
Fmax = 6π ≈ 18.8495559215
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Additional comment
The minimum slope of sin(3x) is the minimum of -3cos(3x), which is -3. This is perfectly balanced by the added 3x term in f(x), so the minimum slope of f(x) is 0 where 3x is an odd multiple of π. Then f(x) is increasing everywhere except at those points where the slope is 0. The graph shows this.