identify the open intervals on which the function is increasing or decreasing. (enter your answers using…

Explanation:

Step1: Find the derivative of the function

The derivative of (y = f(x)=\sin(x)+6) is (f^\prime(x)=\cos(x)) (using the derivative rule ((\sin x)^\prime=\cos x) and ((C)^\prime = 0) for a constant (C)).

Step2: Determine where (f^\prime(x)>0) (increasing)

Set (\cos(x)>0). For (0 < x<2\pi), (\cos(x)>0) when (0 < x<\frac{\pi}{2}) and (\frac{3\pi}{2}<x < 2\pi). In interval notation, the function is increasing on (\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)).

Step3: Determine where (f^\prime(x)<0) (decreasing)

Set (\cos(x)<0). For (0 < x<2\pi), (\cos(x)<0) when (\frac{\pi}{2}<x<\frac{3\pi}{2}). In interval notation, the function is decreasing on (\left(\frac{\pi}{2},\frac{3\pi}{2}\right)).

Answer:

Increasing: (\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)) Decreasing: (\left(\frac{\pi}{2},\frac{3\pi}{2}\right))