Find the monotone increasing interval of function f (x) = 3sin (x / 2 + π / 6) + 3

Find the monotone increasing interval of function f (x) = 3sin (x / 2 + π / 6) + 3

Obviously, when (x / 2 + π / 6) ∈ [- π / 2 + 2K π, π / 2 + 2K π] (k is an integer), sin (x / 2 + π / 6) increases monotonically, → (x / 2) ∈ [- π / 2 - π / 6 + 2K π, π / 2 - π / 6 + 2K π] = [- 2 π / 3 + 2K π, π / 3 + 2K π], → x ∈ [- 4 π / 3 + 4K π, 2 π / 3 + 4K π] (k is an integer)

If f (x) = 3sin (2x + π / 3 + φ), π belongs to (0, π), and f (/ X /) = f (x)
Finding the value of φ

φ belongs to (0, π)
f(lxl)=f(x)
Explain that f (x) = f (- x)
Even functions,
Trigonometric functions are even functions and cosine functions,
And ∵ 3sin (2x + π / 2) = 3cos2x
So there's Phi + π / 3 = π / 2
φ=π/6
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Given the function f (x) = 3sin (ω x + ψ), G (x) = 3cos (ω x + ψ), if for any x ∈ R, f (π / 6 + x) = f (π / 6-x), then G (π / 6) =?

From F (π / 6 + x) = f (π / 6-x), we can see that f (x) is even function with respect to x = π / 6
Then sin (x π / 6 + ψ) = 1 = sin (K π / 2) (k is an integer)
Then x π / 6 + ψ = k π / 2 (k is an integer)
g(π/6)=3cos(xπ/6+Ψ)=3cos(kπ/2)=0

The known function f (x) = 2Sin (2x - π / 6), X ∈ R
1. Write out the equation of the axis of symmetry of function f (x), the coordinates of the center of symmetry and the monotone interval
2. Find the maximum and minimum of function f (x) in the interval [0, π / 2]

1. Axis of symmetry x = π / 3 + K π / 2
Center of symmetry (π / 12 + K π / 2,0)
It increases monotonically on (- π / 6 + K π / 2, π / 3 + K π / 2) and decreases monotonically on (π / 3 + K π / 2, 5 π / 6 + K π / 2)
2. When x = π / 3, max = 2; when x = 0, min = - 1