The increasing interval of the function y = 2Sin (2 π / 3-3x)

The increasing interval of the function y = 2Sin (2 π / 3-3x)

Increasing (2k-1 / 2) π

Function f (x) = lgkx − 1 X − 1 (K ∈ R, and K ＞ 0) (1) Find the definition domain of function (2) If the function f (x) increases monotonically on [10, + ∞), find the value range of K

(1) K > 0 and KX − 1
x−1＞0．
When 0 < K < 1, the definition domain is {x| x < 1 or x > 1
k} When k = 1, the definition field is {x | x ≠ 1}; when k > 1, the definition field is {x | 1 or x < 1
k}；
(2) ∵ the function f (x) increases monotonically on [10, + ∞),
∴y=kx−1
x−1=k+k−1
X − 1 increases monotonically on [10, + ∞) and is positive,
ν k-1 < 0 and 10K − 1
10−1＞0，
∴1
10＜k＜1．

The known function f (x) = 2Sin (KX / 5 + π / 3) (K ≠ 0) Try to find the minimum positive integer k, when the independent variable x changes between any two integers (including the integer itself), at least one value of function f (x) is the maximum value M and the minimum value N The answer is k = 32 why?

To ensure a maximum value and a minimum value is to ensure that the function image has at least one complete waveform between any two integers, that is, a period, and the minimum interval between any two integers is 1, so as long as the function period T

The function f (x) = 2Sin (x - π / 3) + 1, if the period of the function y = f (KX) (k > 0) is 2 π / 3, when x ∈ [0, π / 3], the equation f (KX) = m has exactly two differences For the same solution, find the value range of real number m?

(KX) = 2Sin (KX - π / 3) + 1t = 2 π / k = 2 π / k = 2 π / 3K = 3Y = f (KX) = 2Sin (3x - π / 3) + 1x ∈ [0, π / 3] 3x - π / 3 ∈ [- π / 3,2 π / 3,] sin (3x - π / 3) ∈ [- π / 3,2 π / 3,] sin (3x - π / 3) ∈ [- √ 3 / 2,1] y = f (KX) = 2Sin (3x - π / 3) + 1) ∈ [1 - √ 3,3] when the [1 - √ 3,3] is, when the [1 - √ 3,3] is [1 - [1 - whenx = π / 3, y = √ 3 + 1m ∈ [1 + √ 3,3)

If the period of the function f (x) = 2Sin (KX + Pai / 3) is t, and t belongs to (1,3), then the positive integer k is

The period of F (x) = 2Sin (KX + π / 3)
T=2π/|k|
If t ∈ (1,3) and K is a positive integer, then
T=2π/k∈(1,3)
1/k∈(1/(2π),3/(2π))
k∈((2π)/3,2π)
The K can be 3,4,5,6
Have a good time!

Given the function f (x) = 2Sin (KX / 3 + π / 4), if the period of F (x) is within (2 / 3,3 / 4), find the value of positive integer K

T=2π/ω
=2π/(k/3)
=6πk
From question t ∈ (2 / 3,3 / 4)
K ∈ (8 π, 9 π) is obtained
3.14*8=25.12
3.14*9=28.26
∴k∈{26,27,28}
The character is hard to find

Given the function y = 2Sin (2x + π / 3), we can find that: 1. Amplitude, period and initial phase. 2. Find his equation of symmetry axis and monotone increasing interval

1 a = 2 T = 2 π / 2 = π initial phase = π / 3 222x + π / 3 = π / 2 + K π, so the axis of symmetry is x = π / 12 + K π / 2 - π / 2 + 2K π ≤ 2x + π / 3 ≤ π / 2 + 2K π, so - 5 π / 12 + K π ≤ x ≤ π / 12 + K π, so the increasing range is [- 5 π / 12 + K π, π / 12 + K π]

Known function y = 2Sin (π / 3-2x) Find its symmetry axis equation Find its monotone increasing interval

1. The symmetry axis of sine function is the point where the function gets the minimum or maximum value
So the equation of symmetry axis is as follows:
π / 3-2x = π / 2 + 2K π or π / 3-2x = - π / 2 + 2K π
It is concluded that x = - π / 12-k π or x = 5 π / 12-k π, where k is an arbitrary integer
2
-π/2+2kπ

For the function y = 2Sin (2x + π / 3) + 1 (1) Find the domain of definition and range of value (2) find the minimum positive period of a function (3) find the monotone increasing interval of a function (4) if x ∈ [- π / 4, π / 4], find the range of value of the function (5) write the equation of symmetry axis and the center of symmetry of the function. Find out the super detailed solution process of each small problem

(1) X can be any real number, so the definition domain is R: (- ∞, + ∞); when the sine function sin (2x + π / 3) takes the maximum or minimum value, the corresponding function y takes the maximum or minimum value; therefore, the maximum f (x) = 2 * 1 + 1 = 3, the minimum f (x) = 2 * (- 1) + 1 = - 1; that is, the range is [- 1,3]; (2) the minimum positive period T =

Known function y=2sin (2x- π /3) +3 The maximum value and minimum value of the function are obtained, and the set of the maximum value and minimum value is x is obtained, and the monotone interval of the function is obtained

When sin (2x - π / 3) = 1,2x - π / 3 = π / 2 + 2K π, x = 5 π / 12 + K π
When sin (2x - π / 3) = - 1,2x - π / 3 = - π / 2 + 2K π, x = - π / 12 + K π is obtained
Single increase - π /2+2K π＜ 2x- π /3 ＜π /2+2K π to find the X range
Simply subtract π / 2 + 2K π < 2x - π / 3 < 3 π / 2 + 2K π to find the X range
Analogy sine function, look at 2x - π / 3 as a whole