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

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Chapter 3, 3.3 - Problem 16 - Calculus of a Single Variable (10th Edition, Ron Larson).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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This function is differentiable and it is increasing when its derivative is positive (and decreasing when derivative is negative). f'(x)=2sin(x)cos(x) + cos(x)=cos(x)*(1+2sin(x)). This is zero where cos(x)=0 or sin(x)=-1/2. Such points are pi/2, (7pi)/6, (3pi)/2 and (11pi)/6. We know the signs of the factors, cos(x) and 1+2sin(x), and can determine sign of f'. f' is positive and f is increasing on (0, pi/2), ((7pi)/6, (3pi)/2), ((11pi)/6, 2pi). f is decreasing on (pi/2, (7pi)/6) and ((3pi)/2, (11pi)/6).

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