Find the open intervals on which the function is increasing or decreasing. Apply the First Derivative Test to identify the relative extrema. f(x)=x-2sin(x)

Answer 1

Taking the first derivative with respect to x of the function described yields the following:

d/dx[f(x)] = d/dx[x-2sin(x)] = 1 - 2cos x

Solving for local extrema by solving for the zeros of this derivative yields:

0 = 1 - 2cos(x)

1 = 2cos(x)

1/2 = cos(x)

x = invcos(1/2) = 2Pi*k - Pi/3, 2Pi*k + Pi/3, where k is an integer.

We now have our zeros, the boundaries of our regions. Inputting test values into the derivative determines whether these are regions of increase or decrease.

f'(-Pi/2) = 1 - 2*cos(-Pi/2) = 1 - 2*0 = 1, increasing region

f'(0) = 1 - 2*cos(0) = 1 - 2*(1) = -1, decreasing region

f'(Pi/2) = 1 - 2*cos(Pi/2) = 1 - 2*0 = 1, increasing region

Decreasing domain (2*Pi*k - Pi/3, 2*Pi*k + Pi/3), where k is an integer

Increasing domain (2*Pi*k + Pi/3, 2*Pi*(k+1) - Pi/3), where k is an integer.