What is the extreme f(x)=6sin^2(x)+6sin(x) ?

The extreme f(x)=6sin^2(x)+6sin(x) is

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extreme f(x)=6sin^2(x)+6sin(x)

extreme

Maximum (\frac{π}{2} + 2 πn, 12), Minimum (\frac{7 π}{6} + 2 πn, - \frac{3}{2}), Maximum (\frac{3 π}{2} + 2 πn, 0), Minimum (\frac{11 π}{6} + 2 πn, - \frac{3}{2})

Solution steps

Find the critical points:

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{11 π}{6} + 2 πn

Domain of

6 sin^{2} (x) + 6 sin (x) : - \infty < x < \infty

Find intervals:

Increasing

: 2 πn \leq x < \frac{π}{2} + 2 πn,

Decreasing

: \frac{π}{2} + 2 πn < x < \frac{7 π}{6} + 2 πn,

Increasing

: \frac{7 π}{6} + 2 πn < x < \frac{3 π}{2} + 2 πn,

Decreasing

: \frac{3 π}{2} + 2 πn < x < \frac{11 π}{6} + 2 πn,

Increasing

: \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{2} + 2 πn

into

6 sin^{2} (x) + 6 sin (x) \Rightarrow y = 12

Maximum (\frac{π}{2} + 2 πn, 12)

Plug the extreme point

x = \frac{7 π}{6} + 2 πn

into

6 sin^{2} (x) + 6 sin (x) \Rightarrow y = - \frac{3}{2}

Minimum (\frac{7 π}{6} + 2 πn, - \frac{3}{2})

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

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

Maximum (\frac{3 π}{2} + 2 πn, 0)

Plug the extreme point

x = \frac{11 π}{6} + 2 πn

into

6 sin^{2} (x) + 6 sin (x) \Rightarrow y = - \frac{3}{2}

Minimum (\frac{11 π}{6} + 2 πn, - \frac{3}{2})

Maximum (\frac{π}{2} + 2 πn, 12), Minimum (\frac{7 π}{6} + 2 πn, - \frac{3}{2}), Maximum (\frac{3 π}{2} + 2 πn, 0), Minimum (\frac{11 π}{6} + 2 πn, - \frac{3}{2})

What is the extreme f(x)=6sin^2(x)+6sin(x) ?
The extreme f(x)=6sin^2(x)+6sin(x) is