What is the extreme f(x)=x-2sin(x),(0,2pi) ?

The extreme f(x)=x-2sin(x),(0,2pi) is Minimum(pi/3 , pi/3-sqrt(3)),Maximum((5pi)/3 ,(5pi)/3+sqrt(3))

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extreme f(x)=x-2sin(x),(0,2pi)

extreme

Minimum (\frac{π}{3}, \frac{π}{3} - 3), Maximum (\frac{5 π}{3}, \frac{5 π}{3} + 3)

Solution steps

Find the critical points:

x = \frac{π}{3}, x = \frac{5 π}{3}

Domain of

x - 2 sin (x), (0, 2 π) : 0 < x < 2 π

Find intervals:

Decreasing

: 0 < x < \frac{π}{3},

Increasing

: \frac{π}{3} < x < \frac{5 π}{3},

Decreasing

: \frac{5 π}{3} < x < 2 π

Plug the extreme point

x = \frac{π}{3}

into

x - 2 sin (x) \Rightarrow y = \frac{π}{3} - 3

Minimum (\frac{π}{3}, \frac{π}{3} - 3)

Plug the extreme point

x = \frac{5 π}{3}

into

x - 2 sin (x) \Rightarrow y = \frac{5 π}{3} + 3

Maximum (\frac{5 π}{3}, \frac{5 π}{3} + 3)

Minimum (\frac{π}{3}, \frac{π}{3} - 3), Maximum (\frac{5 π}{3}, \frac{5 π}{3} + 3)

What is the extreme f(x)=x-2sin(x),(0,2pi) ?
The extreme f(x)=x-2sin(x),(0,2pi) is Minimum(pi/3 , pi/3-sqrt(3)),Maximum((5pi)/3 ,(5pi)/3+sqrt(3))