# `f(x) = sin(x) - 1 , 0 < x < 2pi` Identify the open intervals on which the function is increasing or decreasing.

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Expert Answers

embizze | Certified Educator

Given f(x)=sin(x)-1:

A function is increasing on an interval if it is differentiable on the interval and the first derivative is positive; decreasing if the derivative is negative.

f'(x)=cos(x)

The cosine function is positive from (-pi/2,pi/2) with a period of 2pi.

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f(x)=sin(x)-1 is increasing on the open intervals (-pi/2 +/- 2npi,pi/2 +/- 2npi) where n is an integer.

f(x) is decreasing on (pi/2 +/- 2npi,3pi/2 +/- 2npi) for n an integer.

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The graph: