What is the extreme x-2sin(x)(0.2pi) ?

The extreme x-2sin(x)(0.2pi) is

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extreme x-2sin(x)(0.2pi)

extreme

Minimum (arccos (0.79577 \dots) + 2 πn, 0.65051 \dots + 6.28318 \dots n - 1.25663 \dots sin (0.65051 \dots + 6.28318 \dots n)), Maximum (2 π - arccos (0.79577 \dots) + 2 πn, 6.28318 \dots n + 5.63267 \dots - 1.25663 \dots sin (6.28318 \dots n + 5.63267 \dots))

Solution steps

Find the critical points:

x = arccos (0.79577 \dots) + 2 πn, x = 2 π - arccos (0.79577 \dots) + 2 πn

Domain of

x - 2 sin (x) 0.2 π : - \infty < x < \infty

Find intervals:

Decreasing

: 6.28318 \dots n \leq x < arccos (0.79577 \dots) + 2 πn,

Increasing

: arccos (0.79577 \dots) + 2 πn < x < 2 π - arccos (0.79577 \dots) + 2 πn,

Decreasing

: 2 π - arccos (0.79577 \dots) + 2 πn < x < 6.28318 \dots + 6.28318 \dots n

Plug the extreme point

x = arccos (0.79577 \dots) + 2 πn

into

x - 2 sin (x) 0.2 π \Rightarrow y = 0.65051 \dots + 6.28318 \dots n - 1.25663 \dots sin (0.65051 \dots + 6.28318 \dots n)

Minimum (arccos (0.79577 \dots) + 2 πn, 0.65051 \dots + 6.28318 \dots n - 1.25663 \dots sin (0.65051 \dots + 6.28318 \dots n))

Plug the extreme point

x = 2 π - arccos (0.79577 \dots) + 2 πn

into

x - 2 sin (x) 0.2 π \Rightarrow y = 6.28318 \dots n + 5.63267 \dots - 1.25663 \dots sin (6.28318 \dots n + 5.63267 \dots)

Maximum (2 π - arccos (0.79577 \dots) + 2 πn, 6.28318 \dots n + 5.63267 \dots - 1.25663 \dots sin (6.28318 \dots n + 5.63267 \dots))

Minimum (arccos (0.79577 \dots) + 2 πn, 0.65051 \dots + 6.28318 \dots n - 1.25663 \dots sin (0.65051 \dots + 6.28318 \dots n)), Maximum (2 π - arccos (0.79577 \dots) + 2 πn, 6.28318 \dots n + 5.63267 \dots - 1.25663 \dots sin (6.28318 \dots n + 5.63267 \dots))