What is the extreme-sin(x-pi/6) ?

The extreme-sin(x-pi/6) is Minimum((2pi)/3+2pin,-1),Maximum((5pi)/3+2pin,1)

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extreme-sin(x-pi/6)

extreme points

- sin (x - \frac{π}{6})

Minimum (\frac{2 π}{3} + 2 πn, - 1), Maximum (\frac{5 π}{3} + 2 πn, 1)

Solution steps

Find the critical points:

x = \frac{2 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Domain of

- sin (x - \frac{π}{6}) : - \infty < x < \infty

Find intervals:

Decreasing

: 2 πn \leq x < \frac{2 π}{3} + 2 πn,

Increasing

: \frac{2 π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn,

Decreasing

: \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{2 π}{3} + 2 πn

into

- sin (x - \frac{π}{6}) \Rightarrow y = - 1

Minimum (\frac{2 π}{3} + 2 πn, - 1)

Plug the extreme point

x = \frac{5 π}{3} + 2 πn

into

- sin (x - \frac{π}{6}) \Rightarrow y = 1

Maximum (\frac{5 π}{3} + 2 πn, 1)

Minimum (\frac{2 π}{3} + 2 πn, - 1), Maximum (\frac{5 π}{3} + 2 πn, 1)

What is the extreme-sin(x-pi/6) ?
The extreme-sin(x-pi/6) is Minimum((2pi)/3+2pin,-1),Maximum((5pi)/3+2pin,1)