What is the extreme f(x)=4θ-6sin(θ) ?

The extreme f(x)=4θ-6sin(θ) is

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extreme f(x)=4θ-6sin(θ)

extreme

Minimum (arccos (\frac{2}{3}) + 2 πn, 4 (arccos (\frac{2}{3}) + 2 πn) - 6 sin (arccos (\frac{2}{3}) + 2 πn)), Maximum (2 π - arccos (\frac{2}{3}) + 2 πn, 4 (2 π - arccos (\frac{2}{3}) + 2 πn) - 6 sin (2 π - arccos (\frac{2}{3}) + 2 πn))

Solution steps

Find the critical points:

θ = arccos (\frac{2}{3}) + 2 πn, θ = 2 π - arccos (\frac{2}{3}) + 2 πn

Domain of

4 θ - 6 sin (θ) : - \infty < θ < \infty

Find intervals:

Decreasing

: 2 πn \leq θ < arccos (\frac{2}{3}) + 2 πn,

Increasing

: arccos (\frac{2}{3}) + 2 πn < θ < 2 π - arccos (\frac{2}{3}) + 2 πn,

Decreasing

: 2 π - arccos (\frac{2}{3}) + 2 πn < θ < 2 π + 2 πn

Plug the extreme point

x = arccos (\frac{2}{3}) + 2 πn

into

4 θ - 6 sin (θ) \Rightarrow y = 4 (arccos (\frac{2}{3}) + 2 πn) - 6 sin (arccos (\frac{2}{3}) + 2 πn)

Minimum (arccos (\frac{2}{3}) + 2 πn, 4 (arccos (\frac{2}{3}) + 2 πn) - 6 sin (arccos (\frac{2}{3}) + 2 πn))

Plug the extreme point

x = 2 π - arccos (\frac{2}{3}) + 2 πn

into

4 θ - 6 sin (θ) \Rightarrow y = 4 (2 π - arccos (\frac{2}{3}) + 2 πn) - 6 sin (2 π - arccos (\frac{2}{3}) + 2 πn)

Maximum (2 π - arccos (\frac{2}{3}) + 2 πn, 4 (2 π - arccos (\frac{2}{3}) + 2 πn) - 6 sin (2 π - arccos (\frac{2}{3}) + 2 πn))

Minimum (arccos (\frac{2}{3}) + 2 πn, 4 (arccos (\frac{2}{3}) + 2 πn) - 6 sin (arccos (\frac{2}{3}) + 2 πn)), Maximum (2 π - arccos (\frac{2}{3}) + 2 πn, 4 (2 π - arccos (\frac{2}{3}) + 2 πn) - 6 sin (2 π - arccos (\frac{2}{3}) + 2 πn))