What is the extreme 6sqrt(3cos(x)+6sin^2(x)) ?

The extreme 6sqrt(3cos(x)+6sin^2(x)) is

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extreme 6sqrt(3cos(x)+6sin^2(x))

extreme

Minimum (2 πn, 6 3 cos (2 πn) + 6 sin^{2} (2 πn)), Saddle (arccos (\frac{1}{4}) + 2 πn, 6 3 cos (arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (arccos (\frac{1}{4}) + 2 πn)), Maximum (arccos (\frac{- 17 + 1}{4}), 0), Maximum (- arccos (\frac{- 17 + 1}{4}) + 2 π, 0), Saddle (2 π - arccos (\frac{1}{4}) + 2 πn, 6 3 cos (2 π - arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (2 π - arccos (\frac{1}{4}) + 2 πn))

Solution steps

Find the critical points:

x = 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Domain of

6 3 cos (x) + 6 sin^{2} (x) : - arccos (\frac{- 17 + 1}{4}) + 2 πn \leq x \leq arccos (\frac{- 17 + 1}{4}) + 2 πn

Find intervals:

Increasing

: 2 πn < x < arccos (\frac{1}{4}) + 2 πn,

Increasing

: arccos (\frac{1}{4}) + 2 πn < x < 2 π + 2 πn,

Decreasing

: 2 πn \leq x < 2 π - arccos (\frac{1}{4}) + 2 πn,

Decreasing

: 2 π - arccos (\frac{1}{4}) + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = 2 πn

into

6 3 cos (x) + 6 sin^{2} (x) \Rightarrow y = 6 3 cos (2 πn) + 6 sin^{2} (2 πn)

Minimum (2 πn, 6 3 cos (2 πn) + 6 sin^{2} (2 πn))

Plug the extreme point

x = arccos (\frac{1}{4}) + 2 πn

into

6 3 cos (x) + 6 sin^{2} (x) \Rightarrow y = 6 3 cos (arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (arccos (\frac{1}{4}) + 2 πn)

Saddle (arccos (\frac{1}{4}) + 2 πn, 6 3 cos (arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (arccos (\frac{1}{4}) + 2 πn))

Plug the extreme point

x = arccos (\frac{- 17 + 1}{4})

into

6 3 cos (x) + 6 sin^{2} (x) \Rightarrow y = 0

Maximum (arccos (\frac{- 17 + 1}{4}), 0)

Plug the extreme point

x = - arccos (\frac{- 17 + 1}{4}) + 2 π

into

6 3 cos (x) + 6 sin^{2} (x) \Rightarrow y = 0

Maximum (- arccos (\frac{- 17 + 1}{4}) + 2 π, 0)

Plug the extreme point

x = 2 π - arccos (\frac{1}{4}) + 2 πn

into

6 3 cos (x) + 6 sin^{2} (x) \Rightarrow y = 6 3 cos (2 π - arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (2 π - arccos (\frac{1}{4}) + 2 πn)

Saddle (2 π - arccos (\frac{1}{4}) + 2 πn, 6 3 cos (2 π - arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (2 π - arccos (\frac{1}{4}) + 2 πn))

Minimum (2 πn, 6 3 cos (2 πn) + 6 sin^{2} (2 πn)), Saddle (arccos (\frac{1}{4}) + 2 πn, 6 3 cos (arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (arccos (\frac{1}{4}) + 2 πn)), Maximum (arccos (\frac{- 17 + 1}{4}), 0), Maximum (- arccos (\frac{- 17 + 1}{4}) + 2 π, 0), Saddle (2 π - arccos (\frac{1}{4}) + 2 πn, 6 3 cos (2 π - arccos (\frac{1}{4}) + 2 πn) + 6 sin^{2} (2 π - arccos (\frac{1}{4}) + 2 πn))

What is the extreme 6sqrt(3cos(x)+6sin^2(x)) ?
The extreme 6sqrt(3cos(x)+6sin^2(x)) is